f(t) = 3400(1.001)^52t, with t in years. Rewrite it with a yearly factor, to five decimal places, and give the yearly rate of growth as a percent, to the nearest hundredth.

f(t) is about 3400(1.05335)^t; 5.33% growth a year

f(t) = 4200(0.9925)^12t, with t in years. Rewrite it with a yearly factor, to five decimal places, and give the yearly rate of decay as a percent, to the nearest hundredth.

f(t) is about 4200(0.91362)^t; 8.64% decay a year

f(t) = 5000(1.025)^4t, with t in years. Rewrite it with a yearly factor, to five decimal places, and give the yearly rate of growth as a percent, to the nearest hundredth.

f(t) is about 5000(1.10381)^t; 10.38% growth a year

f(t) = 2800(1.12)^t, with t in years. Rewrite it with a monthly factor, to five decimal places.

f(t) is about 2800(1.00949)^12t

Worksheets · Precalculus

Exponential and logarithmic models worksheet

Logarithmic scales squeeze huge ranges into small numbers: one step on the Richter scale is ten times the amplitude, and 10 decibels is ten times the intensity. Exponential models run the other way, from a rate to a value.

Select your difficulty

Pick more than one for a sheet that mixes them.

The preview can look cramped on a phone. The PDF and printed sheet come out normal.



Sigma Prep · sigmaprep.io/worksheets Name Exponential and Log Models Date Period Answer each question. Use the pH formula $p H = - ⁣ \log [H^{+}]$ . How many times as acidic, by hydrogen ion concentration, is a solution of pH 5 as one of pH 9? $p H = - ⁣ \log [H^{+}]$ . A base is added to a solution of pH 9 until its hydrogen ion concentration is one ten-thousandth of what it was. What is its new pH? $p H = - ⁣ \log [H^{+}]$ . Find the pH when $[H^{+}] = 7.2 \times 10^{- 10}$ , to the nearest hundredth. $p H = - ⁣ \log [H^{+}]$ . Solution A has pH 5, and solution B has 10,000 times the hydrogen ion concentration of A. What is the pH of B? Use the decibel formula $L = 10 \log \frac{I}{I_{0}}$ . A sound measures 40 dB. Another is 1,000 times as intense. What is its level? One sound is 100 times as intense as another. How many decibels louder is it? One sound is 60 dB louder than another. How many times as intense is it? One sound is 10,000 times as intense as another. How many decibels louder is it? Use the Richter scale On the Richter scale $M = \log \frac{A}{A_{0}}$ . Two earthquakes measure 2.8 and 4.8. The larger one's seismograph amplitude is how many times the smaller one's? On the Richter scale $M = \log \frac{A}{A_{0}}$ . An earthquake of magnitude 7.0 is followed by one whose amplitude is 10 times as large. What is the magnitude of the second? On the Richter scale $M = \log \frac{A}{A_{0}}$ . How many times as large is the amplitude of a magnitude 7.5 earthquake as one of magnitude 4.5? Use Newton's law of cooling An $84^{\circ} C$ cup of coffee cools in a $23^{\circ} C$ room. After 8 minutes it is $44^{\circ} C.$ When is it $33^{\circ} C,$ to the nearest tenth of a minute? An $80^{\circ} C$ cup of tea cools in a $16^{\circ} C$ room. After 12 minutes it is $46^{\circ} C.$ Find $k$ in $T (t) = 16 + 64 e^{- k t}$ , to three decimal places. A $90^{\circ} C$ bowl of soup cools in a $17^{\circ} C$ room. After 8 minutes it is $68^{\circ} C.$ Find $k$ in $T (t) = 17 + 73 e^{- k t}$ , to three decimal places. Evaluate a logistic model $P (t) = \frac{450}{1 + 14 e^{- 0.4 t}}$ . Find $P (5)$ , to the nearest tenth. $P (t) = \frac{300}{1 + 5 e^{- 0.25 t}}$ . Find $P (7)$ , to the nearest tenth. $P (t) = \frac{300}{1 + 4 e^{- 0.4 t}}$ . Find $P (6)$ , to the nearest tenth. Rewrite with a yearly factor and give the yearly rate $f (t) = 3400 {(1.001)}^{52 t}$ , with $t$ in years. Rewrite it with a yearly factor, to five decimal places, and give the yearly rate of growth as a percent, to the nearest hundredth. $f (t) = 4200 {(0.9925)}^{12 t}$ , with $t$ in years. Rewrite it with a yearly factor, to five decimal places, and give the yearly rate of decay as a percent, to the nearest hundredth. $f (t) = 5000 {(1.025)}^{4 t}$ , with $t$ in years. Rewrite it with a yearly factor, to five decimal places, and give the yearly rate of growth as a percent, to the nearest hundredth. Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Exponential and Log Models

Date Period

Answer each question.

Use the pH formula
$p H = - ⁣ \log [H^{+}]$ . How many times as acidic, by hydrogen ion concentration, is a solution of pH 5 as one of pH 9?
$p H = - ⁣ \log [H^{+}]$ . A base is added to a solution of pH 9 until its hydrogen ion concentration is one ten-thousandth of what it was. What is its new pH?
$p H = - ⁣ \log [H^{+}]$ . Find the pH when $[H^{+}] = 7.2 \times 10^{- 10}$ , to the nearest hundredth.
$p H = - ⁣ \log [H^{+}]$ . Solution A has pH 5, and solution B has 10,000 times the hydrogen ion concentration of A. What is the pH of B?

Use the decibel formula
$L = 10 \log \frac{I}{I_{0}}$ . A sound measures 40 dB. Another is 1,000 times as intense. What is its level?
One sound is 100 times as intense as another. How many decibels louder is it?
One sound is 60 dB louder than another. How many times as intense is it?
One sound is 10,000 times as intense as another. How many decibels louder is it?

Use the Richter scale
On the Richter scale $M = \log \frac{A}{A_{0}}$ . Two earthquakes measure 2.8 and 4.8. The larger one's seismograph amplitude is how many times the smaller one's?
On the Richter scale $M = \log \frac{A}{A_{0}}$ . An earthquake of magnitude 7.0 is followed by one whose amplitude is 10 times as large. What is the magnitude of the second?
On the Richter scale $M = \log \frac{A}{A_{0}}$ . How many times as large is the amplitude of a magnitude 7.5 earthquake as one of magnitude 4.5?

Use Newton's law of cooling
An $84^{\circ} C$ cup of coffee cools in a $23^{\circ} C$ room. After 8 minutes it is $44^{\circ} C.$ When is it $33^{\circ} C,$ to the nearest tenth of a minute?
An $80^{\circ} C$ cup of tea cools in a $16^{\circ} C$ room. After 12 minutes it is $46^{\circ} C.$ Find $k$ in $T (t) = 16 + 64 e^{- k t}$ , to three decimal places.
A $90^{\circ} C$ bowl of soup cools in a $17^{\circ} C$ room. After 8 minutes it is $68^{\circ} C.$ Find $k$ in $T (t) = 17 + 73 e^{- k t}$ , to three decimal places.

Evaluate a logistic model
$P (t) = \frac{450}{1 + 14 e^{- 0.4 t}}$ . Find $P (5)$ , to the nearest tenth.
$P (t) = \frac{300}{1 + 5 e^{- 0.25 t}}$ . Find $P (7)$ , to the nearest tenth.
$P (t) = \frac{300}{1 + 4 e^{- 0.4 t}}$ . Find $P (6)$ , to the nearest tenth.

Rewrite with a yearly factor and give the yearly rate
$f (t) = 3400 {(1.001)}^{52 t}$ , with $t$ in years. Rewrite it with a yearly factor, to five decimal places, and give the yearly rate of growth as a percent, to the nearest hundredth.
$f (t) = 4200 {(0.9925)}^{12 t}$ , with $t$ in years. Rewrite it with a yearly factor, to five decimal places, and give the yearly rate of decay as a percent, to the nearest hundredth.
$f (t) = 5000 {(1.025)}^{4 t}$ , with $t$ in years. Rewrite it with a yearly factor, to five decimal places, and give the yearly rate of growth as a percent, to the nearest hundredth.



Sigma Prep · sigmaprep.io/worksheets Name Exponential and Log ModelsAnswer keyVersion 1 Date Period Answer each question. Use the pH formula $p H = - ⁣ \log [H^{+}]$ . How many times as acidic, by hydrogen ion concentration, is a solution of pH 5 as one of pH 9? $10^{4}$ $p H = - ⁣ \log [H^{+}]$ . A base is added to a solution of pH 9 until its hydrogen ion concentration is one ten-thousandth of what it was. What is its new pH? $13$ $p H = - ⁣ \log [H^{+}]$ . Find the pH when $[H^{+}] = 7.2 \times 10^{- 10}$ , to the nearest hundredth. $9.14$ $p H = - ⁣ \log [H^{+}]$ . Solution A has pH 5, and solution B has 10,000 times the hydrogen ion concentration of A. What is the pH of B? $1$ Use the decibel formula $L = 10 \log \frac{I}{I_{0}}$ . A sound measures 40 dB. Another is 1,000 times as intense. What is its level? $70$ dB One sound is 100 times as intense as another. How many decibels louder is it? $20$ dB One sound is 60 dB louder than another. How many times as intense is it? $10^{6}$ One sound is 10,000 times as intense as another. How many decibels louder is it? $40$ dB Use the Richter scale On the Richter scale $M = \log \frac{A}{A_{0}}$ . Two earthquakes measure 2.8 and 4.8. The larger one's seismograph amplitude is how many times the smaller one's? $10^{2}$ On the Richter scale $M = \log \frac{A}{A_{0}}$ . An earthquake of magnitude 7.0 is followed by one whose amplitude is 10 times as large. What is the magnitude of the second? $8.0$ On the Richter scale $M = \log \frac{A}{A_{0}}$ . How many times as large is the amplitude of a magnitude 7.5 earthquake as one of magnitude 4.5? $10^{3}$ Use Newton's law of cooling An $84^{\circ} C$ cup of coffee cools in a $23^{\circ} C$ room. After 8 minutes it is $44^{\circ} C.$ When is it $33^{\circ} C,$ to the nearest tenth of a minute? $13.6$ minutes An $80^{\circ} C$ cup of tea cools in a $16^{\circ} C$ room. After 12 minutes it is $46^{\circ} C.$ Find $k$ in $T (t) = 16 + 64 e^{- k t}$ , to three decimal places. $0.063$ A $90^{\circ} C$ bowl of soup cools in a $17^{\circ} C$ room. After 8 minutes it is $68^{\circ} C.$ Find $k$ in $T (t) = 17 + 73 e^{- k t}$ , to three decimal places. $0.045$ Evaluate a logistic model $P (t) = \frac{450}{1 + 14 e^{- 0.4 t}}$ . Find $P (5)$ , to the nearest tenth. $155.5$ $P (t) = \frac{300}{1 + 5 e^{- 0.25 t}}$ . Find $P (7)$ , to the nearest tenth. $160.5$ $P (t) = \frac{300}{1 + 4 e^{- 0.4 t}}$ . Find $P (6)$ , to the nearest tenth. $220.1$ Rewrite with a yearly factor and give the yearly rate $f (t) = 3400 {(1.001)}^{52 t}$ , with $t$ in years. Rewrite it with a yearly factor, to five decimal places, and give the yearly rate of growth as a percent, to the nearest hundredth. $f (t) \approx 3400 {(1.05335)}^{t}$ ; $5.33 %$ growth a year $f (t) = 4200 {(0.9925)}^{12 t}$ , with $t$ in years. Rewrite it with a yearly factor, to five decimal places, and give the yearly rate of decay as a percent, to the nearest hundredth. $f (t) \approx 4200 {(0.91362)}^{t}$ ; $8.64 %$ decay a year $f (t) = 5000 {(1.025)}^{4 t}$ , with $t$ in years. Rewrite it with a yearly factor, to five decimal places, and give the yearly rate of growth as a percent, to the nearest hundredth. $f (t) \approx 5000 {(1.10381)}^{t}$ ; $10.38 %$ growth a year Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Exponential and Log ModelsAnswer keyVersion 1

Date Period

Answer each question.

Use the pH formula
$p H = - ⁣ \log [H^{+}]$ . How many times as acidic, by hydrogen ion concentration, is a solution of pH 5 as one of pH 9? $10^{4}$
$p H = - ⁣ \log [H^{+}]$ . A base is added to a solution of pH 9 until its hydrogen ion concentration is one ten-thousandth of what it was. What is its new pH? $13$
$p H = - ⁣ \log [H^{+}]$ . Find the pH when $[H^{+}] = 7.2 \times 10^{- 10}$ , to the nearest hundredth. $9.14$
$p H = - ⁣ \log [H^{+}]$ . Solution A has pH 5, and solution B has 10,000 times the hydrogen ion concentration of A. What is the pH of B? $1$

Use the decibel formula
$L = 10 \log \frac{I}{I_{0}}$ . A sound measures 40 dB. Another is 1,000 times as intense. What is its level? $70$ dB
One sound is 100 times as intense as another. How many decibels louder is it? $20$ dB
One sound is 60 dB louder than another. How many times as intense is it? $10^{6}$
One sound is 10,000 times as intense as another. How many decibels louder is it? $40$ dB

Use the Richter scale
On the Richter scale $M = \log \frac{A}{A_{0}}$ . Two earthquakes measure 2.8 and 4.8. The larger one's seismograph amplitude is how many times the smaller one's? $10^{2}$
On the Richter scale $M = \log \frac{A}{A_{0}}$ . An earthquake of magnitude 7.0 is followed by one whose amplitude is 10 times as large. What is the magnitude of the second? $8.0$
On the Richter scale $M = \log \frac{A}{A_{0}}$ . How many times as large is the amplitude of a magnitude 7.5 earthquake as one of magnitude 4.5? $10^{3}$

Use Newton's law of cooling
An $84^{\circ} C$ cup of coffee cools in a $23^{\circ} C$ room. After 8 minutes it is $44^{\circ} C.$ When is it $33^{\circ} C,$ to the nearest tenth of a minute? $13.6$ minutes
An $80^{\circ} C$ cup of tea cools in a $16^{\circ} C$ room. After 12 minutes it is $46^{\circ} C.$ Find $k$ in $T (t) = 16 + 64 e^{- k t}$ , to three decimal places. $0.063$
A $90^{\circ} C$ bowl of soup cools in a $17^{\circ} C$ room. After 8 minutes it is $68^{\circ} C.$ Find $k$ in $T (t) = 17 + 73 e^{- k t}$ , to three decimal places. $0.045$

Evaluate a logistic model
$P (t) = \frac{450}{1 + 14 e^{- 0.4 t}}$ . Find $P (5)$ , to the nearest tenth. $155.5$
$P (t) = \frac{300}{1 + 5 e^{- 0.25 t}}$ . Find $P (7)$ , to the nearest tenth. $160.5$
$P (t) = \frac{300}{1 + 4 e^{- 0.4 t}}$ . Find $P (6)$ , to the nearest tenth. $220.1$

Rewrite with a yearly factor and give the yearly rate
$f (t) = 3400 {(1.001)}^{52 t}$ , with $t$ in years. Rewrite it with a yearly factor, to five decimal places, and give the yearly rate of growth as a percent, to the nearest hundredth. $f (t) \approx 3400 {(1.05335)}^{t}$ ; $5.33 %$ growth a year
$f (t) = 4200 {(0.9925)}^{12 t}$ , with $t$ in years. Rewrite it with a yearly factor, to five decimal places, and give the yearly rate of decay as a percent, to the nearest hundredth. $f (t) \approx 4200 {(0.91362)}^{t}$ ; $8.64 %$ decay a year
$f (t) = 5000 {(1.025)}^{4 t}$ , with $t$ in years. Rewrite it with a yearly factor, to five decimal places, and give the yearly rate of growth as a percent, to the nearest hundredth. $f (t) \approx 5000 {(1.10381)}^{t}$ ; $10.38 %$ growth a year

How to do these

$p H = - ⁣ \log [H^{+}]$ . Find the pH when $[H^{+}] = 3.2 \times 10^{- 5}$

log(3.2 × 10⁻⁵) is log 3.2 minus 5.
log 3.2 is about 0.505, so the log is about -4.495.
Change the sign.

The answer is $4.49$ .

Where students go wrong

Reading a difference on a log scale as a difference in size. A magnitude 7 earthquake is not a little bigger than a 6: its amplitude is ten times as large.

Also called logarithmic scales, exponential models or log models.

What you can put on this worksheet

pH: $p H = - ⁣ \log [H^{+}]$ . A solution has $[H^{+}] = 0.00001$ . What is its pH? → $5$
Decibels: $L = 10 \log \frac{I}{I_{0}}$ . A sound measures 80 dB. How many times as intense as $I_{0}$ is it? → $10^{8}$
The Richter scale: On the Richter scale $M = \log \frac{A}{A_{0}}$ . An earthquake of magnitude 5.5 is followed by one whose amplitude is 100 times as large. What is the magnitude of the second? → $7.5$
Newton's law of cooling: An $81^{\circ} C$ mug of cocoa cools in a $23^{\circ} C$ room. By Newton's law of cooling its temperature is $T (t) = 23 + 58 e^{- 0.05 t}$ after $t$ minutes. What is it after 14 minutes, to the nearest tenth of a degree? → ${51.8}^{\circ}$ C
Evaluate the solution: $P (t) = \frac{200}{1 + 7 e^{- 0.2 t}}$ . Find $P (6)$ , to the nearest tenth. → $64.3$
The same growth per month or quarter: $f (t) = 2800 {(1.12)}^{t}$ , with $t$ in years. Rewrite it with a monthly factor, to five decimal places. → $f (t) \approx 2800 {(1.00949)}^{12 t}$

Questions about these worksheets

Yes. Every exponential and logarithmic models sheet is free to print and free to download as a PDF. There is no account to make, no email to hand over and no limit on how many you take.

Yes. The answer key prints on a second page, and you can choose whether it shows each question beside its answer or just the answers in a list.

Yes. Print straight from the page, or download the PDF and print that. The sheet is laid out for paper rather than squeezed off a screen, so there is room to work under each question, and the answer key comes out on its own page.

Precalculus is usually taken around 11th or 12th grade, though schools vary and plenty of students meet it earlier or later. Pick the difficulty rather than the grade: the easy level suits a first lesson on it, the hard level suits review before a test.

Yes, as many as you like. The questions are built fresh each time rather than picked from a fixed set of files, so pressing Generate gives a new sheet. That is what you want for a second class, a retake, or two students sitting next to each other.

That is the closest comparison, and Kuta Software is good software. It is also paid software you install on a computer. These worksheets run in a browser tab for free, with no account and nothing to install. Like Kuta, the exponential and logarithmic models questions are generated when you ask for them rather than pulled from a fixed set of files, so two classes never get the same sheet, and the answer key comes with it.

Yes. There are three levels, and you can tick more than one for a sheet that mixes them, in which case the questions come out easiest first. The harder levels are not just larger numbers: they ask for something the easy ones do not.

You pick from 6: ph, decibels, the richter scale, newton's law of cooling, evaluate the solution and the same growth per month or quarter. Tick as many as you want and set how many of each, or let it spread them evenly.

Reading a difference on a log scale as a difference in size. A magnitude 7 earthquake is not a little bigger than a 6: its amplitude is ten times as large.

Exponential and logarithmic models worksheet

Select your difficulty

Use the pH formula

Use the decibel formula

Use the Richter scale

Use Newton's law of cooling

Evaluate a logistic model

Rewrite with a yearly factor and give the yearly rate

Use the pH formula

Use the decibel formula

Use the Richter scale

Use Newton's law of cooling

Evaluate a logistic model

Rewrite with a yearly factor and give the yearly rate

How to do these

Where students go wrong

What you can put on this worksheet

Questions about these worksheets

Are these exponential and logarithmic models worksheets free?+

Do the exponential and logarithmic models worksheets come with an answer key?+

Are these exponential and logarithmic models worksheets printable?+

What grade are these exponential and logarithmic models worksheets for?+

Can I get a different version of the same exponential and logarithmic models worksheet?+

Is this a free alternative to Kuta Software for exponential and logarithmic models?+

Can I choose how hard the exponential and logarithmic models questions are?+

What kinds of question are on a exponential and logarithmic models worksheet?+

What do students get wrong with exponential and logarithmic models?+