Teacher Guide: Logarithm Word Problems

Use logarithms to solve real problems: pH, decibels, earthquake magnitudes, doubling time, half-life and how long money takes to grow.

Present LessonView Student Page

Use this lesson with your class

Free, no student accounts needed.

Share with students

Students open the lesson and practise with instant feedback.

Printable worksheet

All practice problems on paper, with a separate answer key.

Worksheet Answer key

Class quiz

10 questions on Logarithms. Students join with a name, you see everyone's score.

For Teachers

Learning Objectives

Calculate pH, decibel levels and Richter comparisons with the common logarithm
Convert back from a logarithmic scale using powers of 10
Solve growth and decay equations for time by taking a logarithm of both sides
Find doubling times and half-life times and interpret them in context
Round answers as instructed and decide when a situation needs whole periods

Prerequisites

• Common Logarithms (Base 10)
• Natural Logarithms (ln)
• Power Rule for Logarithms
• Solving Logarithmic Equations
• Exponential growth and decay models, including compound interest

Discussion Starters

1. Why do scientists use a logarithmic scale for sound instead of measuring intensity directly?
2. A news report says an earthquake was "only one point stronger". Is that a small difference?
3. Your calculator gives 14.21 years. Why might the real answer be 15 years?
4. How could you check a half-life answer without logarithms?
5. Where else might a quantity grow by the same percentage each period?

Common Misconceptions

A logarithmic scale behaves like a normal scale (pH 4 is "twice as acidic" as pH 8)

Remediation: Build a table: pH 8, 7, 6, 5, 4 with $[H^+] = 10^{-8}, \dots, 10^{-4}$ . Students see each step is a factor of 10, so pH 4 vs pH 8 is $10^4 = 10\,000$ times.

Taking the log of only one side of an equation, or "cancelling" the base

Remediation: Stress that an equation stays balanced only if both sides get the same operation. Write $\ln(\text{left}) = \ln(\text{right})$ explicitly before using the power rule.

The decimal answer is always the final answer

Remediation: Give contexts with whole periods (yearly interest, daily doses) and ask students to test the years just below and above their decimal answer.

Differentiation Ideas

For Struggling Students:

• Start with powers of 10 only (pH of $10^{-3}$ , $10^6$ times the threshold) so no calculator is needed
• Give a fill-in template: formula, isolate the power, take a logarithm, round
• Use the interactive graph to read approximate times before computing them

For On-Level Students:

• Mix pH, decibel, Richter and interest problems with different rounding instructions
• Compare the rule of 72 with exact doubling times for several rates
• Solve half-life problems where the answer is not a whole number of half-lives

For Advanced Students:

• Derive the formula $t = \frac{\ln(A/P)}{\ln(1+r)}$ in general and explain why any base works
• Compare yearly, monthly and continuous compounding times to reach the same goal
• Investigate earthquake energy, which grows by a factor of about $10^{1.5} \approx 31.6$ per magnitude

Standards Alignment

HSF-LE.A.4 (CCSS.MATH.CONTENT.HSF.LE.A.4)
For exponential models, express as a logarithm the solution to $ab^{ct} = d$ and evaluate it using technology
HSA-CED.A.1 (CCSS.MATH.CONTENT.HSA.CED.A.1)
Create equations in one variable and use them to solve problems, including exponential functions
HSN-Q.A.3 (CCSS.MATH.CONTENT.HSN.Q.A.3)
Choose a level of accuracy appropriate to limitations on measurement when reporting quantities
HSF-BF.B.5 (CCSS.MATH.CONTENT.HSF.BF.B.5)
Understand the inverse relationship between exponents and logarithms and use it to solve problems

Lesson Resources

visualDoubling Time Graph
Graph of $y = 1000 \cdot 1.05^x$ and $y = 2000$ : the intersection shows the doubling time $14.21$ years
visualLogarithmic Number Line
Powers of 10 spaced evenly, like the decibel scale: each step is $10$ dB more
visualpH Scale as Powers of 10
Hydrogen ion concentrations from $10^{-14}$ to $1$ mol/L spaced evenly: each step to the right means $10$ times more $[H^+]$ and a pH one unit lower
activityKitchen pH Lab
Students look up the pH of vinegar, milk, soap and lemon juice and compute how many times more acidic one is than another
worksheetLogarithm Word Problems
Mixed practice: pH, decibels, earthquakes, compound interest and half-life, with rounding instructions

Lesson Content

Everything students see: definition, examples, common mistakes, applications. Tap to open.

Definition

A logarithm word problem describes a real situation in which the unknown is an exponent: a time, a magnitude or a level. You solve it in three moves: write the formula, isolate the power, then take a logarithm of both sides.

Two kinds of problems appear again and again:

1. Logarithmic scales (the formula already contains a logarithm):

pH of a solution: $\text{pH} = -\log[H^+]$ , where $[H^+]$ is the hydrogen ion concentration in mol/L
Sound level in decibels: $L = 10 \log\left(\frac{I}{I_0}\right)$ , where $I_0$ is the threshold of hearing
Richter magnitude: two earthquakes whose magnitudes differ by $d$ have amplitudes in the ratio $10^{d}$

2. Growth and decay over time (the unknown

t

sits in the exponent):

A = P(1 + r)^t \quad\Rightarrow\quad t = \frac{\ln\left(\frac{A}{P}\right)}{\ln(1 + r)}

For continuous growth

A = Pe^{rt}

gives

t = \frac{\ln(A/P)}{r}

, and for a half-life

h

the amount

N = N_0 \left(\frac{1}{2}\right)^{t/h}

gives

t = h \cdot \frac{\ln(N/N_0)}{\ln(0.5)}

Here

\log

means the common logarithm (base 10) and

\ln

the natural logarithm (base

e

). Most answers are decimals, so every problem states how to round, and you round only at the very end.

Worked Examples

A soft drink has a hydrogen ion concentration of $[H^+] = 3.2 \times 10^{-4}$ mol/L. Find its pH, rounded to two decimal places.

Write the formula

$\text{pH} = -\log[H^+] = -\log(0.00032)$ → $\text{pH} = -\log(0.00032)$

Evaluate the common logarithm

$\log(0.00032) \approx -3.49485$

Apply the minus sign

$\text{pH} = -(-3.49485) = 3.49485$

Round to two decimal places

The third decimal is $4$ , so round down → $\text{pH} \approx 3.49$

Answer: $\text{pH} \approx 3.49$ (acidic, because it is below 7)

Common Mistakes

Saying a magnitude 8 earthquake is "2 times" or "twice" as strong as a magnitude 6 earthquake

Why it's wrong: The Richter scale is logarithmic: each whole step multiplies the amplitude by 10. Differences in magnitude are exponents, not multipliers.

Correct: Ratio of amplitudes $= 10^{8 - 6} = 10^2 = 100$ times.

Forgetting the minus sign in $\text{pH} = -\log[H^+]$ and getting a negative pH like $-3.49$

Why it's wrong: Concentrations such as $0.00032$ are less than 1, so their logarithm is negative. The minus sign in the formula makes the pH positive.

Correct: $\text{pH} = -\log(0.00032) = -(-3.49485) \approx 3.49$

Using the rate instead of the growth factor: $t = \frac{\ln 2}{\ln 0.05}$

Why it's wrong: The balance is multiplied by $1 + r = 1.05$ each year, not by $0.05$ . Also $\ln 0.05$ is negative, which would give a negative time.

Correct: $t = \frac{\ln 2}{\ln(1.05)} \approx 14.21$ years. For decay use $1 - r$ , for example $0.85$ for a $15\%$ loss.

Rounding too early, for example writing $\ln 1.05 \approx 0.05$

Why it's wrong: Small roundings inside a quotient grow: $\frac{0.6931}{0.05} \approx 13.86$ instead of $14.21$ , which is wrong by about 4 months.

Correct: Keep full calculator precision (or at least 5 significant digits) and round only the final answer, as the problem says.

Why It Matters

Logarithms turn huge ranges of numbers into small, readable ones, and they are the only tool that pulls an unknown out of an exponent:

Chemistry: lemon juice (pH about 2) has $100\,000$ times more hydrogen ions than pure water (pH 7).
Health: a 110 dB concert is $1000$ times more intense than an 80 dB street, which is why hearing protection matters.
Geology: a magnitude 7 earthquake shakes with $100$ times the amplitude of a magnitude 5 one.
Money: banks, savers and loan officers ask "how many years until my money doubles?" and answer it with $\ln$ .
Medicine: doctors use half-life to decide when a medicine has dropped below an effective level.

Every one of these questions is an exponent question, and a logarithm answers it.

Real World Applications

Acid Rain

Environmental scientists compare the acidity of rain with pH. Because pH is a logarithm, a small pH difference means a big difference in hydrogen ions.

Example:

Normal rain has a pH of about $5.6$ . Rain with pH $4.6$ has $10^{1} = 10$ times as many hydrogen ions.

1Try It Yourself

Normal rain has a pH of $5.6$ . A storm brings acid rain with a pH of $4.3$ .

How many times more hydrogen ions does the acid rain contain? Round to the nearest whole number.

Step 1: Write the mathematical expression

Compute $10^{5.6 - 4.3}$ and round to the nearest whole number:

Hearing Protection at Concerts

Sound engineers and audiologists work in decibels. Ear plugs are rated by how much they cut the intensity of sound.

Example:

Dividing the intensity by $10$ lowers the level by $10$ dB, dividing by $100$ lowers it by $20$ dB.

2Try It Yourself

A rock concert measures $110$ dB. Ear plugs reduce the sound intensity reaching your ear to $\frac{1}{1000}$ of its value.

What sound level, in decibels, reaches your ear? (The answer is a whole number, no rounding needed.)

Step 1: Write the mathematical expression

Write the new level as $10 \log\left(\frac{10^{11}}{1000}\right)$ and evaluate:

Planning a Savings Goal

Financial planners use logarithms to find how many years an investment needs to reach a goal.

Example:

At $4\%$ per year, money grows by the factor $1.04$ each year, so the number of years is $t = \frac{\ln(\text{goal}/\text{start})}{\ln 1.04}$ .

3Try It Yourself

You invest 5000 dollars at $4\%$ interest compounded once a year. Your goal is 8000 dollars.

How many full years must you wait until the balance first reaches the goal?

Step 1: Write the mathematical expression

Solve $5000 \cdot 1.04^t = 8000$ , then round up to whole years:

Key Takeaways

1In a logarithm word problem the unknown is an exponent: a time, a magnitude or a level
2pH: $\text{pH} = -\log[H^+]$ ; going back: $[H^+] = 10^{-\text{pH}}$
3Decibels: $L = 10\log\left(\frac{I}{I_0}\right)$ ; every $10$ dB more means $10$ times the intensity
4Richter: magnitudes that differ by $d$ mean amplitudes in the ratio $10^{d}$
5Time in growth or decay: isolate the power, take $\ln$ of both sides, then $t = \frac{\ln(A/P)}{\ln(1 \pm r)}$ ; doubling time $\frac{\ln 2}{\ln(1+r)}$
6Round only at the end, exactly as the problem asks, and check if the situation needs whole years

Frequently Asked Questions

Should I use log or ln to solve for time?

Either one works, as long as you use the same one on top and bottom:

\frac{\log 2}{\log 1.05} = \frac{\ln 2}{\ln 1.05} \approx 14.21

. Formulas for pH and decibels are defined with the common logarithm

\log

(base 10), so use

\log

there.

Why is the answer 15 years if the calculator says 14.21?

The equation treats time as continuous, but a bank adds interest only once a year. After 14 years the balance is still below the target, so you need the 15th payment. Always ask whether the situation allows fractions of a year.

Can a pH be negative or greater than 14?

Mathematically yes:

[H^+] = 2

mol/L gives

\text{pH} = -\log 2 \approx -0.30

. In everyday chemistry, though, almost all solutions fall between 0 and 14.

Is there a quick estimate for doubling time?

The "rule of 72" says doubling time is about

72 \div (\text{rate in }\%)

. At

5\%

72 \div 5 = 14.4

years, close to the exact

14.21

. Use logarithms when the problem asks for an exact answer.

Glossary

pH: A measure of acidity: $\text{pH} = -\log[H^+]$ . Below 7 is acidic, 7 is neutral, above 7 is basic.
Decibel (dB): Unit of sound level: $L = 10\log\left(\frac{I}{I_0}\right)$ , where $I_0$ is the threshold of hearing.
Richter scale: A logarithmic scale for earthquakes: one whole step means $10$ times the amplitude of the ground motion.
Doubling time: The time a growing quantity needs to double: $t = \frac{\ln 2}{\ln(1 + r)}$ for growth by $r$ per period.
Half-life: The time a decaying quantity needs to fall to half of its value.
Growth factor: The number a quantity is multiplied by each period: $1 + r$ for growth, $1 - r$ for decay.

Formula Card

pH

\text{pH} = -\log[H^+] \iff [H^+] = 10^{-\text{pH}}

Acidity from the hydrogen ion concentration

Sound level

L = 10\log\left(\frac{I}{I_0}\right) \iff \frac{I}{I_0} = 10^{L/10}

Decibels from the intensity ratio

Richter comparison

\frac{A_1}{A_2} = 10^{M_1 - M_2}

Ratio of amplitudes of two earthquakes

Time to reach a target

t = \frac{\ln(A/P)}{\ln(1 + r)}

From $A = P(1 + r)^t$; use $1 - r$ for decay

Continuous growth

t = \frac{\ln(A/P)}{r}

From $A = Pe^{rt}$

Half-life

t = h \cdot \frac{\ln(N/N_0)}{\ln 0.5}

From $N = N_0 \cdot 0.5^{t/h}$ with half-life $h$

Teacher Guide: Logarithm Word Problems

Use this lesson with your class

For Teachers

For Struggling Students:

For On-Level Students:

For Advanced Students:

Lesson Content

Definition

Worked Examples

Common Mistakes

Why It Matters

Real World Applications

Acid Rain

Hearing Protection at Concerts

Planning a Savings Goal

Key Takeaways

Frequently Asked Questions

Glossary

Formula Card

pH

Sound level

Richter comparison

Time to reach a target

Continuous growth

Half-life

More in This Topic

Introduction to Logarithms

Definition of a Logarithm

Common Logarithms (Base 10)

Natural Logarithms (ln)