Precalculus Exponential and Logarithmic Functions Worksheet with Answers – PDF

Precalculus Exponential Log Worksheet: problems 1–5

1. Exponential to logarithmic form

Write each exponential equation in logarithmic form.

1. \(2^{7}=128\)
2. \(10^{-3}=0.001\)
3. \(9^{1/2}=3\)
4. \(e^{0}=1\)

2. Logarithmic to exponential form

Write each logarithmic equation in exponential form.

1. \(\log_{5} 125=3\)
2. \(\log_{4}\frac{1}{64}=-3\)
3. \(\log_{36} 6=\frac{1}{2}\)
4. \(\ln 20=t\)

3. Evaluate logarithms

Evaluate each logarithm without a calculator.

1. \(\log_{3} 81\)
2. \(\log_{2}\frac{1}{32}\)
3. \(\log_{25} 5\)
4. \(\log_{1/4} 16\)
5. \(\ln\sqrt{e}\)

4. Points on an inverse graph

The graph shows \(f(x)=2^{x}\) and the labeled points A, B, C and D.

1. Write the coordinates of the matching points A′, B′, C′ and D′ on the graph of \(f^{-1}\).
2. Write an equation for \(f^{-1}(x)\).
3. State the domain of \(f^{-1}\).

5. Finding inverse functions

Find the inverse function of each function.

1. \(f(x)=4^{x}\)
2. \(f(x)=2\cdot 3^{x}\)
3. \(f(x)=\ln (x+3)\)
4. \(f(x)=1+\log_{2} x\)

Precalculus Exponential Log Worksheet: problems 6–10

6. Verify inverses by composition

Show that \(f(x)=e^{2x}\) and \(g(x)=\frac{1}{2}\ln x\) are inverse functions.

7. Features of a log graph

The graph shows \(g(x)=\log_{3}(x-2)+1\).

1. Give the equation of the vertical asymptote.
2. State the domain and the range.
3. Find the exact x-intercept.
4. Find \(g(29)\).

8. Expand logarithms

Use the laws of logarithms to expand each expression.

1. \(\log_{b}(x^{3}y^{2})\)
2. \(\ln\frac{5x}{y^{4}}\)
3. \(\log_{2}\sqrt{\frac{8x}{z}}\)

9. Condense logarithms

Write each expression as a single logarithm.

1. \(2\log 3+\log 5\)
2. \(\ln 12-\ln 4+2\ln x\)
3. \(\frac{1}{2}\log_{2}x+2\log_{2}y-3\log_{2}z\)

10. Using given logarithm values

Given \(\log 2\approx 0.3010\) and \(\log 3\approx 0.4771\), approximate each value without a calculator key for logarithms.

1. \(\log 6\)
2. \(\log 18\)
3. \(\log 1.5\)
4. \(\log 8\)
5. \(\log\frac{2}{9}\)

Precalculus Exponential Log Worksheet: problems 11–15

11. Change of base

Use the change-of-base formula to approximate each value to three decimal places.

1. \(\log_{5} 40\)
2. \(\log_{7} 3\)
3. \(\log_{2} 100\)

12. Same-base exponential equations

Solve each equation by writing both sides with the same base.

1. \(4^{x+1}=32\)
2. \(9^{x}=27^{x-2}\)
3. \((\frac{1}{2})^{x}=16\)
4. \(25^{x-1}=125^{x}\)

13. Solve using logarithms

Solve each equation. Give an exact answer, then round to three decimal places.

1. \(3\cdot 2^{x}=60\)
2. \(7^{x-1}=50\)
3. \(5e^{2x}=45\)
4. \(12(0.9)^{x}=3\)

14. Estimate, then solve

The graph shows \(y=5e^{0.4x}\).

1. Use the graph to estimate the solution of \(5e^{0.4x}=40\).
2. Solve the equation exactly and round to three decimal places.

15. Logarithmic equations

Solve each equation and reject any extraneous solution.

1. \(\log_{2}(x+3)+\log_{2}(x-3)=4\)
2. \(\ln x+\ln (x-2)=\ln 8\)
3. \(\log_{3}(2x-1)=2\)
4. \(\log (x+6)=2\log x\)

Precalculus Exponential Log Worksheet: problems 16–20

16. Quadratic form

Solve each equation.

1. \(e^{2x}-5e^{x}+6=0\)
2. \(4^{x}-6\cdot 2^{x}+8=0\)

17. Transformations of an exponential

Let \(h(x)=-2^{x-1}+3\). Describe how its graph is related to the graph of \(y=2^{x}\), then answer the questions.

1. Describe the transformations.
2. Give the equation of the horizontal asymptote.
3. Find the y-intercept.
4. Find the exact x-intercept.
5. State the range.

18. Rewrite growth and decay models

Answer each question about exponential models.

1. Write \(A=200(0.8)^{t}\) in the form \(A=Ce^{kt}\) and give \(k\) to four decimal places.
2. The model \(A=50e^{0.07t}\) is continuous. By what percent does \(A\) grow in each full time unit, to two decimal places?
3. Write \(P=300\cdot 2^{t/3}\) in the form \(P=300e^{kt}\) and give \(k\) to four decimal places.

19. Compound interest time

Maya deposits $2,000 in an account paying 4% annual interest compounded quarterly. Use \(A=P(1+\frac{r}{n})^{nt}\) to find how many years it takes for the balance to reach $3,000.

20. Continuous compounding

Jamal invests $5,000 at 3.5% interest compounded continuously, \(A=5000e^{0.035t}\). Find the time for the money to double and the time to reach $12,000. Round to two decimal places.

Precalculus Exponential Log Worksheet: problems 21–24

21. Radioactive decay dating

Carbon-14 has a half-life of 5,730 years, so the fraction left after \(t\) years is \((\frac{1}{2})^{t/5730}\). A bone fragment has 30% of its original carbon-14. Find its age to the nearest year.

22. Logistic population model

The number of students who have heard a rumor at a school of 1,200 is modeled by \(P(t)=\frac{1200}{1+19e^{-0.5t}}\), where \(t\) is in hours.

1. How many students knew the rumor at \(t=0\)?
2. What is the carrying capacity?
3. When did exactly 600 students know the rumor?
4. Find \(P(8)\) to the nearest student.

23. pH calculations

The pH of a solution is \(\text{pH}=-\log [H^{+}]\). Round to two decimal places.

1. A sample of rain has \([H^{+}]=3.2\times 10^{-5}\) mol/L. Find its pH.
2. Seawater has pH 8.2. Find \([H^{+}]\) in scientific notation.
3. How many times greater is the hydrogen ion concentration at pH 3 than at pH 6?

24. Decibels and earthquakes

Sound level is \(L=10\log\frac{I}{I_{0}}\) decibels, where \(I_{0}\) is the faintest audible intensity.

1. How many times as intense as a conversation is a lawn mower?
2. Find \(\frac{I}{I_{0}}\) for a rock concert, in scientific notation.
3. An earthquake of magnitude 7.1 is compared with one of magnitude 5.4. By what factor is its wave amplitude greater? Round to the nearest whole number.

Which problems build the basics of logarithms?

Problem 1 rewrites \(2^{7}=128\) and \(10^{-3}=0.001\) in log form. Problem 2 does the reverse, starting from \(\log_{5}125=3\). Then problem 3 evaluates logs such as \(\log_{2}\frac{1}{32}\) with no calculator, so think “which power of 2?” In problem 4, a graph of \(f(x)=2^{x}\) shows four points, and you read the matching points on the inverse.

How do the law and equation problems differ?

Problems 8 and 9 expand and condense expressions. Problem 11 uses change of base for \(\log_{5}40\). Next, problem 12 solves \(4^{x+1}=32\) by matching bases, while problem 13 needs logs for \(3\cdot 2^{x}=60\). Problem 15 asks you to reject extraneous answers, since \(\log_{2}(x+3)+\log_{2}(x-3)=4\) produces one value that breaks the domain. Problem 16 hides a quadratic inside \(e^{2x}-5e^{x}+6=0\).

What do the model problems add?

Compound interest on $2,000 at 4% in problem 19 compares quarterly steps with the continuous case in problem 20. Carbon-14 dating in problem 21 uses a half-life of 5,730 years. Problem 22 follows a rumor through a school of 1,200 students with a logistic curve. Finally, problems 23 and 24 apply log scales to pH, sound and earthquakes.

Take time to label what each variable means before you substitute.

How should you finish the worksheet?

Substitute every equation answer back into the original, because extraneous solutions hide in log problems. Then group your misses into algebra, log laws and modeling. Afterward, open the key and attempt the quiz.

See every chapter on the Precalculus math page.