logarithm in problem solving

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How to Solve Logarithms

Last Updated: March 17, 2024 Fact Checked

This article was reviewed by Grace Imson, MA . Grace Imson is a math teacher with over 40 years of teaching experience. Grace is currently a math instructor at the City College of San Francisco and was previously in the Math Department at Saint Louis University. She has taught math at the elementary, middle, high school, and college levels. She has an MA in Education, specializing in Administration and Supervision from Saint Louis University. This article has been fact-checked, ensuring the accuracy of any cited facts and confirming the authority of its sources. This article has been viewed 253,608 times.

Logarithms might be intimidating, but solving a logarithm is much simpler once you realize that logarithms are just another way to write out exponential equations. Once you rewrite the logarithm into a more familiar form, you should be able to solve it as you would solve any standard exponential equation.

Before You Begin: Learn to Express a Logarithmic Equation Exponentially [1] X Research source [2] X Research source

• If and only if: b y = x
• b does not equal 1
• In the same equation, y is the exponent and x is the exponential expression that the logarithm is set equal to.

• Example: 1024 = ?

• This could also be written as: 4 5

• Example: 4 5 = 1024

Method One: Solve for X

• log 3 ( x + 5) + 6 - 6 = 10 - 6
• log 3 ( x + 5) = 4

• Comparing this equation to the definition [ y = log b (x) ], you can conclude that: y = 4; b = 3; x = x + 5
• Rewrite the equation so that: b y = x
• 3 4 = x + 5

• 3 * 3 * 3 * 3 = x + 5
• 81 - 5 = x + 5 - 5

• Example: x = 76

Method Two: Solve for X Using the Logarithmic Product Rule [3] X Research source [4] X Research source

• log b (m * n) = log b (m) + log b (n)

• log 4 (x + 6) + log 4 (x) = 2 - log 4 (x) + log 4 (x)
• log 4 (x + 6) + log 4 (x) = 2

• log 4 [(x + 6) * x] = 2
• log 4 (x 2 + 6x) = 2

• Comparing this equation to the definition [ y = log b (x) ], you can conclude that: y = 2; b = 4 ; x = x 2 + 6x
• 4 2 = x 2 + 6x

• 4 * 4 = x 2 + 6x
• 16 = x 2 + 6x
• 16 - 16 = x 2 + 6x - 16
• 0 = x 2 + 6x - 16
• 0 = (x - 2) * (x + 8)
• x = 2; x = -8

• Example: x = 2
• Note that you cannot have a negative solution for a logarithm, so you can discard x - 8 as a solution.

Method Three: Solve for X Using the Logarithmic Quotient Rule [5] X Research source

• log b (m / n) = log b (m) - log b (n)

• log 3 (x + 6) - log 3 (x - 2) = 2 + log 3 (x - 2) - log 3 (x - 2)
• log 3 (x + 6) - log 3 (x - 2) = 2

• log 3 [(x + 6) / (x - 2)] = 2

• Comparing this equation to the definition [ y = log b (x) ], you can conclude that: y = 2; b = 3; x = (x + 6) / (x - 2)
• 3 2 = (x + 6) / (x - 2)

• 3 * 3 = (x + 6) / (x - 2)
• 9 = (x + 6) / (x - 2)
• 9 * (x - 2) = [(x + 6) / (x - 2)] * (x - 2)
• 9x - 18 = x + 6
• 9x - x - 18 + 18 = x - x + 6 + 18
• 8x / 8 = 24 / 8

• Example: x = 3

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• ↑ https://www.wtamu.edu/academic/anns/mps/math/mathlab/col_algebra/col_alg_tut43_logfun.htm#logdef
• ↑ https://www.mathsisfun.com/algebra/logarithms.html
• ↑ https://www.wtamu.edu/academic/anns/mps/math/mathlab/col_algebra/col_alg_tut46_logeq.htm
• ↑ https://www.youtube.com/watch?v=fnhFneOz6n8
• ↑ https://www.wtamu.edu/academic/anns/mps/math/mathlab/col_algebra/col_alg_tut44_logprop.htm

About This Article

To solve a logarithm, start by identifying the base, which is "b" in the equation, the exponent, which is "y," and the exponential expression, which is "x." Then, move the exponential expression to one side of the equation, and apply the exponent to the base by multiplying the base by itself the number of times indicated in the exponent. Finally, rewrite your final answer as an exponential expression. To learn how to solve for "x" in a logarithm, scroll down! Did this summary help you? Yes No

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Logarithmic Functions

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Properties of logarithmic functions

Comparison of exponential function and logarithmic function, practice questions, solving logarithmic functions – explanation & examples.

Logarithms and exponents are two topics in mathematics that are closely related. Therefore it is useful we take a brief review of exponents.

An exponent is a form of writing the repeated multiplication of a number by itself. An exponential function is of the form f (x) = b y , where b > 0 < x and b ≠ 1. The quantity x is the number, b is the base, and y is the exponent or power.

For example , 32 = 2 × 2 × 2 × 2 × 2 = 2 2 .

On the other hand, the logarithmic function is defined as the inverse function of exponentiation. Consider again the exponential function f(x) = b y , where b > 0 < x and b ≠ 1. We can represent this function in logarithmic form as:

y = log b x

Then the logarithmic function is given by;

f(x) = log b x = y, where b is the base, y is the exponent, and x is the argument.

The function f (x) = log b x is read as “log base b of x.” Logarithms are useful in mathematics because they enable us to perform calculations with very large numbers.

How to Solve Logarithmic Functions?

To solve the logarithmic functions, it is important to use exponential functions in the given expression. The natural log or ln is the inverse of e . That means one can undo the other one i.e.

ln (e x ) = x

To solve an equation with logarithm(s), it is important to know their properties.

Properties of logarithmic functions are simply the rules for simplifying logarithms when the inputs are in the form of division, multiplication, or exponents of logarithmic values.

Some of the properties are listed below.

• Product rule

The product rule of logarithm states the logarithm of the product of two numbers having a common base is equal to the sum of individual logarithms.

⟹ log a  (p q) = log a  p + log a  q.

• Quotient rule

The quotient rule of logarithms states that the logarithm of the two numbers’ ratio with the same bases is equal to the difference of each logarithm.

⟹ log a  (p/q) = log a  p – log a q

The power rule of logarithm states that the logarithm of a number with a rational exponent is equal to the product of the exponent and its logarithm.

⟹ log a  (p q ) = q log a p

• Change of Base rule

⟹ log a p = log x p ⋅ log a x

⟹ log q p = log x p / log x q

• Zero Exponent Rule

Other properties of logarithmic functions include:

• The bases of an exponential function and its equivalent logarithmic function are equal.
• The logarithms of a positive number to the base of the same number are equal to 1.

log a  a = 1

• Logarithms of 1 to any base are 0.

log a  1 = 0

• Log a 0 is undefined
• Logarithms of negative numbers are undefined.
• The base of logarithms can never be negative or 1.
• A logarithmic function with base 10is called a common logarithm. Always assume a base of 10 when solving with logarithmic functions without a small subscript for the base.

Whenever you see logarithms in the equation, you always think of how to undo the logarithm to solve the equation. For that, you use an exponential function . Both of these functions are interchangeable.

The following table tells the way of writing and interchanging the exponential functions and logarithmic functions . The third column tells about how to read both the logarithmic functions.

Let’s use these properties to solve a couple of problems involving logarithmic functions.

Rewrite exponential function 7 2 = 49 to its equivalent logarithmic function.

Given 7 2 = 64.

Here, the base = 7, exponent = 2 and the argument = 49. Therefore, 7 2 = 64 in logarithmic function is;

⟹ log 7 49 = 2

Write the logarithmic equivalent of 5 3 = 125.

exponent = 3;

and argument = 125

5 3 = 125 ⟹ log 5 125 =3

Solve for x in log  3  x = 2

log  3  x = 2 3 2  = x ⟹ x = 9

If 2 log x = 4 log 3, then find the value of ‘x’.

2 log x = 4 log 3

Divide each side by 2.

log x = (4 log 3) / 2

log x = 2 log 3

log x = log 3 2

log x = log 9

Find the logarithm of 1024 to the base 2.

1024 = 2 10

log 2 1024 = 10

Find the value of x in log 2 ( x ) = 4

Rewrite the logarithmic function log 2 ( x ) = 4 to exponential form.

Solve for x in the following logarithmic function log 2 (x – 1) = 5.

Solution Rewrite the logarithm in exponential form as;

log 2 (x – 1) = 5 ⟹ x – 1 = 2 5

Now, solve for x in the algebraic equation. ⟹ x – 1 = 32 x = 33

Find the value of x in log x 900 = 2.

Write the logarithm in exponential form as;

Find the square root of both sides of the equation to get;

x = -30 and 30

But since, the base of logarithms can never be negative or 1, therefore, the correct answer is 30.

Solve for x given, log x = log 2 + log 5

Using the product rule Log b  (m n) = log b  m + log b  n we get;

⟹ log 2 + log 5 = log (2 * 5) = Log   (10).

Therefore, x = 10.

Solve log  x  (4x – 3) = 2

Rewrite the logarithm in exponential form to get;

x 2  = 4x – 3

Now, solve the quadratic equation. x 2  = 4x – 3 x 2  – 4x + 3 = 0 (x -1) (x – 3) = 0

Since the base of a logarithm can never be 1, then the only solution is 3.

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• \log _2(x+1)=\log _3(27)
• \ln (x+2)-\ln (x+1)=1
• \ln (x)+\ln (x-1)=\ln (3x+12)
• 4+\log _3(7x)=10
• \ln (10)-\ln (7-x)=\ln (x)
• \log _2(x^2-6x)=3+\log _2(1-x)
• How do you calculate logarithmic equations?
• To solve a logarithmic equations use the esxponents rules to isolate logarithmic expressions with the same base. Set the arguments equal to each other, solve the equation and check your answer.
• What is logarithm equation?
• A logarithmic equation is an equation that involves the logarithm of an expression containing a varaible.
• What are the 3 types of logarithms?
• The three types of logarithms are common logarithms (base 10), natural logarithms (base e), and logarithms with an arbitrary base.
• Is log10 and log the same?
• When there's no base on the log it means the common logarithm which is log base 10.
• What is the inverse of log in math?
• The inverse of a log function is an exponantial.

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• High School Math Solutions – Exponential Equation Calculator Solving exponential equations is pretty straightforward; there are basically two techniques: <ul> If the exponents...

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Logarithmic Word Problems

Log Probs Expo Growth Expo Decay

What are logarithm word problems?

Logarithmic word problems, in my experience, generally involve either evaluating a given logarithmic equation at a given point, or else solving an equation for a given variable; they're pretty straightforward.

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What real-world problems use logarithms?

The classic real-world contexts for logarithm word problems are the measurement of acidity or alkalinity (that is, the measurement of pH), the measurement of sound (in decibels, or dB), and the measurement of earthquake intensity (on the Richter scale), among other uses ( link ).

Note: While log-based word problems are, in my experience, pretty straightforward, their statements tend to be fairly lengthy. Expect to have to plow through an unusual amount of text before they get to the point.

• Chemists define the acidity or alkalinity of a substance according to the formula pH =  −log[H + ] where [H + ] is the hydrogen ion concentration, measured in moles per liter. Solutions with a pH value of less than 7 are acidic; solutions with a pH value of greater than 7 are basic; solutions with a pH of 7 (such as pure water) are neutral.

a) Suppose that you test apple juice and find that the hydrogen ion concentration is [H + ] = 0.0003 . Find the pH value and determine whether the juice is basic or acidic.

b) You test some ammonia and determine the hydrogen ion concentration to be [H + ] = 1.3 × 10 −9 . Find the pH value and determine whether the ammonia is basic or acidic.

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In each case, I need to evaluate the pH function at the given value of [H + ] . In other words, this exercise, despite all the verbiage, is just plug-n-chug.

Since no base is specified, I will assume that the base for this logarithm is 10 , so that this is the so-called "common" log. (I happen to know that 10 is indeed the correct base, but they should have specified.)

a) In the case of the apple juice, the hydrogen ion concentration is [H + ] = 0.0003 , so:

pH = −log[H + ]

= −log[0.0003]

= 3.52287874528...

This value is less than 7 , so the apple juice is acidic.

b) In the case of the ammonia, the hydrogen ion concentration is [H + ] = 1.3 × 10 −9 , so:

= −log[1.3 × 10 −9 ] = 8.88605664769...

This value is more than 7 , so the ammonia is basic.

(a) The juice is acidic with a pH of about 3.5 , and (b) the ammonia is basic with a pH of about 8.9 .

When a logarithm is given without a base being specified, different people in different contexts will assume different bases; either 10 , 2 , or e . Ask now whether or not bases will be specified for all exercises, or if you're going to be expected to "just know" the bases for certain formulas, or if you're supposed to "just assume" that all logs without a specified base have a base of... [find out which one].

• "Loudness" is measured in decibels (abbreviated as dB). The formula for the loudness of a sound is given by dB = 10×log[I ÷ I 0 ] where I 0 is the intensity of "threshold sound", or sound that can barely be perceived. Other sounds are defined in terms of how many times more intense they are than threshold sound. For instance, a cat's purr is about 316 times as intense as threshold sound, for a decibel rating of:

dB = 10×log[I ÷ I 0 ]     = 10×log[ (316 I 0 ) ÷ I 0 ]     = 10×log[ 316 ]     = 24.9968708262...

...about 25 decibels.

Considering that prolonged exposure to sounds above 85 decibels can cause hearing damage or loss, and considering that a gunshot from a .22 rimfire rifle has an intensity of about I = (2.5 × 10 13 )I 0 , should you follow the rules and wear ear protection when practicing at the rifle range?

I need to evaluate the decibel equation at I = (2.5 × 10 13 )I 0 :

dB = 10log [ I ÷ I 0 ]     = 10log[ (2.5 ×10 13 )I 0 ÷ I 0 ]     = 10log[2.5 ×10 13 ]     = 133.979400087...

In other words, the squirrel gun creates a noise level of about 134 decibels. Since this is well above the level at which I can suffer hearing damage,

I should follow the rules and wear ear protection.

• Earthquake intensity is measured by the Richter scale. The formula for the Richter rating of a given quake is given by R = log[ I ÷ I 0  ] where I 0 is the "threshold quake", or movement that can barely be detected, and the intensity I is given in terms of multiples of that threshold intensity.

You have a seismograph set up at home, and see that there was an event while you were out that had an intensity of I = 989I 0 . Given that a heavy truck rumbling by can cause a microquake with a Richter rating of 3 or 3.5 , and "moderate" quakes have a Richter rating of 4 or more, what was likely the event that occurred while you were out?

To determine the probable event, I need to convert the intensity of the mystery quake into a Richter rating by evaluating the Richter function at I = 989I 0 :

R = log[ I ÷ I 0 ]     = log[ 989I 0 ÷ I 0 ]     = log[989]     = 2.9951962916...

A Richter rating of about 3 is not high enough to have been a moderate quake.

The event was probably just a big truck.

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Despite the buzz surrounding it, the prominence of prompt engineering may be fleeting. A more enduring and adaptable skill will keep enabling us to harness the potential of generative AI? It is called problem formulation — the ability to identify, analyze, and delineate problems.

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• Oguz A. Acar is a Chair in Marketing at King’s Business School, King’s College London.

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With an estimated 75,000 people unhoused in Los Angeles County, there’s universal agreement among our readers that the situation in Southern California is unacceptable and deserves urgent action. Where they disagree is on what’s causing the homelessness problem and how to address it.

The last L.A. mayoral election, between Karen Bass and Rick Caruso, reflected that divide. It’s also seen here in the letters of our readers when we asked them to weigh in. What we heard fell broadly into two camps: One with arguments focused on law and order, a traditionally conservative view; the other focused on systemic issues such as economic inequality, a traditionally liberal worldview. Whether city leaders shift toward one side or the other can make a big difference in how our region responds.

— Paul Thornton, letters editor

To the editor: California’s housing shortage and homelessness crisis are what happens when you have politicians who have no idea of economics making laws, which have negative effects that the public must suffer.

Supply and demand is the cornerstone of our capitalist system; even a child understands that.

Democrats are driving California into a third-world state, and they wonder why people are fleeing this place.

Marcus Kourtjian, Northridge

To the editor: Please explain to us readers how it is possible that a place such as Skid Row exists in the wealthiest and most Democratic state in the country.

The very eloquent politicians have been working on this since the dawn of time, with budgets to buy small towns in the Midwest. The super wealthy of L.A. talk the liberal talk, do the spiritual work and eat the $20 avocado toasts.

Yet we read about Skid Row as if it’s OK to have such a place downtown. The dystopian future is here, and it has been here for years.

Michele Castagnetti, Los Angeles

— Marcus Kourtjian, Northridge

To the editor: The incompetence of our civic leaders and institutions, including the media, is why we have been dealing with homelessness for many years.

It started with the failure to recognize the extent that mental illness plays a role. Commonsense observations and input from residents were either ignored or dismissed as prejudicial judgments against what many of your columnists might call a “victimized” community.

This failure was part of the next major failure, which was the inability to categorize different classes of homeless populations and come up with a plan to deal with each one separately. It’s somewhat akin to using the same chemotherapy regardless of the type of cancer.

Encampments point to yet another failure, where politicians use court rulings as “my hands are tied” excuses. For example, one such ruling allows for “rudimentary protection from the elements” for those who sleep on the streets. Common sense, which seems to elude our politicians, dictates that should not result in massive hazmat operations when the encampments are actually cleaned up.

People are getting tired of throwing money at this problem with little to show for it. That Proposition 1 barely passed shows that even the typical L.A. Times reader is slowly starting to allow fiscal responsibility to trump virtue signaling by bond measure.

Jesse Goldbaum, Woodland Hills

We read about Skid Row as if it’s OK to have such a place downtown. The dystopian future is here, and it has been here for years.

— Michele Castagnetti, Los Angeles

To the editor: Near Lawrence Middle School in Chatsworth, there has been a homeless encampment for nearly a year. People there have continually refused shelter, and many live with addiction and mental illness.

Typically, you wouldn’t want this close to a school.

Those of us who have contacted City Hall about this have been told that little can be done until anti-camping and vagrancy laws change. Most of the areas in Chatsworth that have encampments cleared quickly get repopulated after massive cleanups.

The public would like our sidewalks back. We feel unheard and extremely frustrated as a community. We want to feel safe and be able to check our mail after dark without the fear of being robbed or stabbed.

Kristen Mason, Chatsworth

To the editor: I wish our political leaders would remember the rest of us in their decisions. They travel to Sacramento and Washington to solicit more funding for homeless services.

What about all the rest of us who live with the consequences of this crisis? The unaddressed blight on our streets. The dying of trees in our parks. The car break-ins.

Many of us mourn the loss of civic pride, which is clearly not a priority now. It is painful to witness.

Why aren’t our leaders doing more to clean up our streets and sidewalks, to get rid of graffiti, to keep our parks clean, maintained and the trees watered, all of which are dismissed as unimportant “quality of life” issues? We would thrive if they did that.

Many of us who once felt compassion and patience for those who live on the streets are angry because our needs and concerns have been overlooked.

Mary A. Fischer, Echo Park

Many heartfelt attempts by many heartfelt politicians have failed residents, who continue to face tents and trash that block sidewalks, and who fear violence by people wandering down the street.

— Buz Wolf, Big Bear Lake, Calif

To the editor: I live fewer than 13 miles from the greatest humanitarian crisis in our nation.

Almost daily, I drive by encampments near the 110 Freeway in Highland Park. Unhoused people who illegally occupy the sidewalks, parkways and embankment terrify law-abiding residents of the area. For more than three years, I have reached out to several local agencies about this. And until recently, the situation would only get worse.

But last month, because of the nonstop effort of Highland Park residents to document and report the dangerous activity on their street, combined with an L.A. Alliance for Human Rights lawsuit, I witnessed the biggest comprehensive cleanup I’ve seen there in three years.

Litter still covers the embankment. Human waste and drug paraphernalia remain. I am guessing the unhoused people who left this area a disaster will eventually return.

Recently, large numbers of L.A. voters gave their support to progressive candidates who do not believe that homeless people should be held to the same legal standards as those of us with homes. If I, as a housed citizen, smoked fentanyl on your street or screamed threats at you, I would be arrested and held accountable.

If we want to live in a society that holds people accountable for their actions and protects children, in addition to sheltering the unsheltered and treating the sick, we must hold everyone to the same standards of lawful behavior.

I am not criminalizing homelessness. I am criminalizing unlawful behavior that has turned Los Angeles into a mess.

Tori Stover Mordecai, San Marino

To the editor: I think it is fair to say that the city and county administrators of Los Angeles have failed miserably at resolving the homelessness crisis.

Is it fair that the overwhelming majority of us are subjected to this? Which heartfelt politicians are looking out for us?

I recall seeing a picture of a temporary but full-service Quonset hut “city” set up for returning World War II veterans in Griffith Park. They, with their families, were housed there as they waited for affordable homes to be built to accommodate them. Once homes were available, they all moved on.

Why can’t this arrangement be duplicated for all homeless people in the L.A. area? Since this project would provide housing and services, enhanced laws against vagrancy and camping would need to be passed and enforced.

Unfortunately, this would require our current bleeding-heart politicians to step aside and for voters to support more pragmatic officials.

Buz Wolf, Big Bear Lake, Calif.

To the editor: Homelessness will remain intractable until our nation addresses economic inequality. We need moral outrage.

Here in L.A., we construct tiny-home villages with 64 square feet of living space and estates in Bel-Air with 30,000.

I commend the mayor’s efforts, but this kind of discrepancy in living conditions has led to revolutions and land reform. Let’s figure this out before it gets bloody.

Lisa Dieckmann, Los Angeles

To the editor: As rents and house prices go up, take note — someone is raking in all that money. That’s where our government leaders should focus their attention.

As long as real estate investors are asking for higher profits, rents will keep going up, and more people will find themselves unable to afford shelter. It’s the typical corporate strategy: Plunder the environment to gain all the riches, and let the public pay for the aftermath.

We need to end the eviction of people from their homes so landlords and developers can build something more expensive and reap greater profits. Let’s regulate the profits that real estate investors can make.

We get daily information about encampments and the plight of neighborhoods and how much the public is paying to house people. We see it for ourselves. Tell us more about the other end of the scale, where the riches pile up.

Sarah Starr, Los Angeles

This kind of discrepancy in living conditions has led to revolutions and land reform. Let’s figure this out before it gets bloody.

— Lisa Dieckmann, Los Angeles

To the editor: Many years after recovering from homelessness and building a great life, I started working on the streets, first in San Francisco and now in Los Angeles, bringing essentials such as hygiene, laundry, haircuts and clothing to people who had lost their homes.

My return from homelessness happened when I was surrounded by a community that shared my experiences and stayed with me through every mistake I made, building resilience that made it possible for me to thrive.

Community is the one thing that I know works; it has worked in my life, and it has worked for millions of others. It’s the inspiration for everything I do today. Community is the only way we will end homelessness in L.A.

Hundreds of thousands of Angelenos are one paycheck or illness away from being unhoused. According to L.A. County officials, more than 220 people here lose their homes every day. The path back to housing for the hardest-hit has become more deadly and more treacherous.

The agencies tasked with helping these people are overwhelmed. Big agencies can’t do this; they never could.

It will take us, the millions of Angelenos who share this beautiful place, the community groups, faith communities and neighbors, to rise up and say we will no longer accept this suffering. We know the answer is us, but can we set aside our differences and unite to end this?

Paul Asplund, Pasadena

— Paul Asplund, Pasadena

To the editor: So many studies blame the lack of affordable housing — and all housing — as the primary contributor to homelessness. I live in an area of single-family homes and know that zoning protecting this kind of neighborhood must end.

It’s not a choice between skyscrapers and single-family homes. There are many alternatives — look at what Santa Monica is doing with mixed-use, mid-rise residential buildings.

Additionally, we need quality care and sufficient funding for drug and mental health programs. This is a complicated problem that needs a multi-pronged approach and time to solve.

Karin Costello, Santa Monica

To the editor: What drives homelessness? In a nutshell, housing policies that reward real estate investors and developers; the scarcity of transitional housing for institutionalized individuals; consistently low wages; increasing numbers of seniors on low fixed incomes; a dearth of “affordable” housing; and the resulting mental health, substance use and medical challenges.

This is not just my opinion; it’s what the UC San Francisco Benioff Homelessness and Housing Initiative study found last year .

In L.A., Mayor Karen Bass’ Inside Safe initiative aims to bring individuals living in encampments into hotel and motel rooms. However, because of the scarcity of low-income permanent housing (and permanent supportive housing), these people end up living in temporary shelters for years. Temporary shelters have become a revolving door, moving individuals in and out.

Local agencies are not able to provide services and housing to the increasing number of individuals living on the street. So unhoused individuals become chronically homeless and have a much harder time adapting to societal expectations.

Once individuals become unhoused, it is difficult for them to find housing again. Their mental health suffers as a result of the trauma of becoming homeless, and they may begin using substances to help them cope.

The need for “affordable” housing has been proposed for decades. On the surface, Bass’ effort to significantly streamline construction for affordable housing seems like a good idea.

But Bass’ initiative does not provide subsidized housing, prevent displacement of low-income tenants or provide actual “affordable” housing. Rather, it caters to the myth that building housing for people at 80% of local median income will magically lower rents throughout the city.

Thus, Bass’ streamlining effort is another failed housing policy that will not appreciably help to increase housing production for homeless and low-income individuals.

Jane Demian, Los Angeles

The question is not how to reduce the number of homeless, but what is the forecasted growth rate in the homeless population?

— Ronald Stein, Irvine

To the editor: The number of people experiencing homelessness has gone up in Los Angeles, and the city is becoming a very different place from the one in which I grew up.

As a lifelong resident of South Los Angeles, I wish city and county leaders knew how the different initiatives passed by voters have been perceived by residents. We are tired of voting on ballot measures that have seemingly little to no impact.

The only thing I have seen improve the situation is the Inside Safe program, which combines the knowledge of service providers with the authority of government officials to actually get people the help they need.

For years on my way to work, I passed an encampment under the 405 Freeway on Venice Boulevard that grew despite the various ballot measures passed to fund housing and services.

Inside Safe is the only program that truly got people into housing, connected them to services and ensured the area would not become repopulated.

Alyssa Melina, South Los Angeles

To the editor: With the average debt in this country greater than $100,000 per person (across credit cards, mortgages, auto loans and student loans), and with more than half of Americans living paycheck to paycheck, society is facing an unsustainable problem where those “financially challenged” will never pay off their continuously increasing debt.

Thus, with the loss of just one paycheck, there are many millions of people on the verge of joining the growing homeless population.

Ronald Stein, Irvine

More to Read

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Letters to the Editor: L.A.’s anti-camping law isn’t meant to fix homelessness. It’s aimed at quality of life

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I'm the superintendent of Ann Arbor schools. Fixing our budget shortfall will be painful. | Letters

Ann Arbor Public Schools is dedicated to providing world-class learning opportunities for our students, and keeping our families, staff, students and community informed about the financial challenges we face. 

With our nearly 91% graduation rate, amazing staff and students and supportive community, it is no surprise our district has been hailed as one of the best public school systems in the nation.  

That is why we are up to the challenge before us and appreciate the opportunity to detail the factors at play in our financial situation, as highlighted in a recent opinion piece in this newspaper . 

More on Ann Arbor schools budget crisis: Layoffs expected as Ann Arbor Public Schools must cut $25 million from budget

This past November, as part of our regular budget review and audit process, the prior superintendent made the Board of Education aware of a shortfall in our current school year budget. After being named interim superintendent, to thoroughly assess our financial picture, I authorized a comprehensive top-to-bottom review of school finances — and called in a third-party financial expert to ensure we have an accurate picture to proceed on solid footing.  

More on Ann Arbor schools budget crisis: Ann Arbor Public Schools is short $25 million. How did we get here?

As a result of that work, it was determined the district will need to reduce approximately $25 million from the 2024-25 operating budget to comply with State of Michigan and Ann Arbor Public Schools Board of Education requirements. This represents approximately a 10% reduction to our district’s overall budget of more than $300 million. Our team is committed to navigating these challenges while continuing to provide our students with a world-class education. 

Our budget challenges stem from three main historical factors:  

• Over the past four years, student enrollment has decreased by 1,123, which decreased the district’s revenue.  
• An increase in employees over the past decade.  
• Increases in operational costs. 

I worked with our district leaders to identify immediate actions we could take to reduce costs and begin planning long-term solutions to ensure this does not happen again. We know some of these steps will be painful.  

We are reducing central office and administrative staff positions, freezing hiring, conducting in-depth reviews of all central office, district and school budgets for efficiencies and cost savings and renegotiating contracts with vendors to identify cost savings on contracted services.   

This will not solve the shortfall on its own given the magnitude of the budget challenges we face.  

We know this has been a difficult process. We are committed to working together with staff and the AAPS community to ensure a comprehensive district budget plan is developed and implemented in a way that minimizes the impact on classrooms and student learning.  

To build a thoughtful plan that moves us forward in a fiscally responsible way, we are seeking feedback from our valued staff and community members. As we continue to gather this feedback, we are hosting a virtual town hall meeting on Monday, April 15, that will allow us to share information and answer questions. It will be online at 6:30 p.m. Monday at  access.live/annarborpublicschools . We also will host additional in-person venues for staff and community feedback and engagement.  

Throughout this process, our district leadership team has remained steadfast in its commitment to transparency and openness. Together we can, and will, get through this challenging situation as a district and community.  

Jazz Parks  

The writer is interim superintendent of the Ann Arbor Public Schools   

Submit a letter to the editor at freep.com/letters , and we may publish it online and in print.

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To Fight Climate Change, We Need New ‘Political Technologies’

By Peter Coy

Opinion Writer

Science alone won’t stop the planet from overheating. But science coupled with political science just might.

That’s the theme of a new book, “Long Problems: Climate Change and the Challenge of Governing Across Time.” It’s by Thomas Hale, an American political scientist who teaches at the University of Oxford’s Blavatnik School of Government.

Hale argues that people are too quick to throw up their hands because the political will to stop climate change is lacking. For political scientists, he writes, “this is not the end but rather the start of the intellectual challenge.”

Hale has specific ideas for how to change institutions and procedures so that today’s inhabitants of Earth give more consideration to tomorrow’s inhabitants. He calls them, at one point, “political technologies,” a phrase I like.

Long problems such as climate change are ones in which there is a long lag between causes and effects. They are hard to solve, especially with today’s institutions. We don’t act early because we’re uncertain about how big the problem is, and it isn’t as salient as the daily emergencies all around us. Our hesitation gives an opening to obstructionist forces. Today’s decision makers vow to protect the planet for future generations, but the unborn multitudes are mere “shadows” to them, as Hale puts it.

On top of all that, Hale writes, “Institutions created to address the early phase of a long problem struggle to remain useful as the problem’s structure develops over time.” Case in point: The United Nations Framework Convention on Climate Change, which was created in 1992. The original concept was for countries to make binding commitments to fight climate change. As the organization has evolved, though, “nothing is agreed until every country agrees on every point,” Hale writes.

That’s not useful. A better approach is the Paris Agreement of 2015, which went into effect the next year. It allows countries to set their own targets for greenhouse gas reductions while triggering a “norm cascade” that induces them to do more and more. Hale likes the Paris Agreement on the whole, though he says it’s not perfect.

Society has already invented institutions and systems that bring future considerations to the fore, Hale writes. The Congressional Budget Office and similar offices in other countries analyze how new legislation will affect economic growth and government finances in the long run. The bond market assesses whether bond issuers, such as governments, will be able to pay back what they owe. Insurance companies — which Hale doesn’t mention — won’t issue policies unless customers take steps to reduce their risks.

On climate, too, there have been efforts to create institutions and processes that help solve Hale’s long problems. Some governments are requiring business to incorporate the “social cost” of carbon into their decisions. And the Intergovernmental Panel on Climate Change brings together the world’s top experts and issues closely followed reports.

There are many more opportunities for political engineering, Hale writes. He approvingly mentions the Finnish Parliament’s cross-party Committee for the Future and the Finnish Government Report on the Future, which interact. He recommends more experimentation in policymaking — as Chinese leaders put it, crossing the river by feeling for stones.

To get the public and lawmakers thinking more about the future, he endorses Britain’s Climate Change Committee, which he writes “has become a significant political force for the long-term interest,” and similar organizations (some of them not as effective) in Hungary, Israel, Malta, Sweden, Tunisia and the United Arab Emirates.

To insulate long problems from partisan politicking, he recommends the appointment of a trustee to oversee climate decisions, analogous to the way a politically insulated central bank is delegated the authority to conduct monetary policy. The California Air Resources Board is “perhaps the strongest, though still imperfect” example of such an institution in the realm of the environment, he writes. (Hale told me he’s not aware of anything quite like the California agency elsewhere in the world.)

“Long Problems” is a kind of nonfiction counterpart to Kim Stanley Robinson’s science fiction book from 2020, “The Ministry for the Future,” which took seriously the idea that future generations need to be given as much consideration as our own.

Hale is a co-leader of the Net Zero Tracker , which tracks the decarbonization progress of countries and companies, and the Net Zero Regulation and Policy Hub . He told me that he has been involved in helping people at the United Nations prepare for a Summit for the Future, which will be held Sept. 22-23. On the U.N. website is an early draft of a declaration to be issued at the summit, which says among other things that “our conduct today will impact future generations exponentially.”

Anne-Marie Slaughter, the chief executive of the think tank New America who was Hale’s adviser on his doctorate at Princeton, shared a byline with Hale and two of his Oxford colleagues on a policy brief , “Toward a Declaration on Future Generations,” that recommends the U.N. appoint “a special envoy or high commissioner” to be a voice for the future.

I kind of prefer “envoy” because it sounds like the person has literally come from the future.

No one solution will instantly end the political obstacles to fighting climate change. Some of the ideas in Hale’s book may not pan out at all. But I give him credit for focusing on how to solve problems in which the cause and the effect are separated by decades. Getting the “political technology” right is every bit as important as inventing better solar cells, wind turbines and batteries.

Outlook: Oliver Allen

Retail sales rose more than expected in March, but the “booming” pace of growth isn’t likely to last, Oliver Allen, a senior U.S. economist at Pantheon Macroeconomics, wrote in a client note on Monday. “It is hard to see how the strength in consumption can continue for much longer, now that real after-tax income growth has slowed markedly, the bulk of excess savings from earlier in the pandemic has been spent, and a raft of leading indicators point to a marked softening in the labor market,” Allen wrote.

Quote of the Day

“Why do we need to make the rich richer to make them work harder but make the poor poorer for the same purpose?”

— Ha-Joon Chang, “Economics: The User’s Guide” (2014)

Peter Coy is a writer for the Opinion section of The Times, covering economics and business. Email him at [email protected] . @ petercoy

Native teens are nearly 4 times as likely to commit suicide. Tribes suing Meta, Google, TikTok, and Snapchat say social media addiction fuels the problem.

• Two tribal nations are suing Meta, Google, TikTok, and Snapchat over youth mental health.
• Native teenagers are up to four times as likely to die by suicide than other groups.
• The Spirit Lake Tribe and the Menominee Indian Tribe say the companies have preyed on vulnerable youth.

Two tribes are suing social media giants, accusing them of contributing to the high suicide rates among Native teenagers by purposely getting kids hooked on their platforms.

The lawsuits were filed in Los Angeles County Superior Court on Tuesday. One was brought on behalf of the Spirit Lake Tribe in North Dakota, and the Menominee Indian Tribe of Wisconsin brought the other. The lawsuits name Facebook, Instagram, Snapchat, TikTok, YouTube, and their parent companies, including Meta and Google as defendants.

"Given historic teen suicide and mental health issues on our Reservation and across Indian Country, Native youth are particularly vulnerable to the negative long-term effects resulting from the intentional, profit-driven design choices being made by these social media platforms," Lonna Jackson-Street, chairperson of Spirit Lake Nation, said in a statement provided to Business Insider.

The lawsuit said suicide rates for tribal youth in the US are three and a half to four times higher than other racial or ethnic groups, according to the Center for Native American Youth.

"Soaring suicide and mental illness have devastated Tribal communities and have pushed already chronically underfunded mental health programs to the breaking point," the lawsuits said.

The lawsuits allege the addictive nature of the platforms has contributed to the youth mental health crisis, including among kids on reservations.

The lawsuits said the social-media companies "deliberately tweaked the design and operation of their apps to exploit the psychology and neurophysiology of kids" and that the companies preyed on an already vulnerable group.

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Among other things, the lawsuits are seeking "equitable relief to fund prevention education and treatment for excessive and problematic use of social media."

"Enough is enough. Endless scrolling is rewiring our teenagers' brains. We are demanding these social media corporations take responsibility for intentionally creating dangerous features that ramp up the compulsive use of social media by the youth on our Reservation," Gena Kakkak, chairwoman of the Menominee Indian Tribe, said in a statement.

In a statement provided to BI, Google spokesperson José Castaneda said: "Providing young people with a safer, healthier experience has always been core to our work. In collaboration with youth, mental health, and parenting experts, we built services and policies to provide young people with age-appropriate experiences and parents with robust controls. The allegations in these complaints are simply not true."

Meta, TikTok, and Snapchat did not immediately respond to requests for comment from Business Insider.

Other lawsuits have been brought over social media addiction, including by dozens of state attorneys general who sued Meta last year. However, these are the first lawsuits over social media addiction brought by federally recognized tribes, according to Robins Kaplan, the firm that filed the suits.

"These social media giants have generated hundreds of billions of dollars in revenue, our complaint alleges, using a growth-at-all-costs strategy that comes at the expense of Native children and teenagers and the Tribal Nations they are part of," Tim Purdon, chair of the firm's American Indian Law and Policy Group and lead counsel to the tribes, said in a statement.

New York City also filed a lawsuit against Facebook, Instagram, TikTok, Snapchat, and YouTube in February, alleging the addictive nature of their platforms was contributing to the youth mental health crisis.

In response, Snap Inc., the parent company of Snapchat, said in a statement to the Associated Press that their app "opens directly to a camera — rather than a feed of content that encourages passive scrolling — and has no traditional public likes or comments."

The statement continued: "While we will always have more work to do, we feel good about the role Snapchat plays in helping close friends feel connected, happy, and prepared as they face the many challenges of adolescence."

A TikTok spokesperson told AP at the time: "TikTok has industry-leading safeguards to support teens' well-being, including age-restricted features, parental controls, an automatic 60-minute time limit for users under 18, and more."

Meta also said the company wants "teens to have safe, age-appropriate experiences online" and that they have dedicated features aimed at helping kids and parents.

On February 28, Axel Springer, Business Insider's parent company, joined 31 other media groups and filed a $2.3 billion suit against Google in Dutch court, alleging losses suffered due to the company's advertising practices.

Watch: TikTok could be banned in US after House vote

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