Training Exponentials & Logarithms Placement Test Practice — Exponentials & Logs
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Placement Test Practice — Exponentials & Logs

25 min Exponentials & Logarithms

Placement Test Practice — Exponentials & Logarithms

This practice test covers exponential functions, logarithms, their properties, and equation-solving techniques from the entire module. Treat it as a diagnostic — work through all the problems under timed conditions and then check your answers.

The problems span the full range of difficulty, from evaluating simple logarithms to solving applied growth-and-decay problems.

Complete this practice test to prepare for placement exam questions on exponential and logarithmic topics.

Practice Test — 25 Questions

1. Evaluate $\log_4(64)$
2. Solve $2^{x} = 128$
3. Expand $\log\!\left(\dfrac{x^2 y}{z^3}\right)$
4. Evaluate $\ln(e^{-3})$
5. Solve $\log(x - 4) = 2$
6. Condense $3\log(a) + \frac{1}{2}\log(b)$
7. Solve $e^x = 100$
8. Evaluate $\log_{27}(9)$
9. If $\$3{,}000$ is invested at 5% compounded annually, find the balance after 6 years.
10. Solve $5^{2x-1} = 25$
11. Evaluate $10^{\log(5) + \log(6)}$
12. Solve $\log_3(2x - 5) = 3$
13. A radioactive material decays by 4% per year. After how many years is half remaining? (Half-life)
14. Simplify $e^{2\ln(3)}$
15. Solve $\log(x) + \log(x + 21) = 2$
16. Convert $\log_8(x) = \frac{2}{3}$ to exponential form and solve.
17. A population triples every 5 years. Starting at 1,000, write the model. Find the population after 15 years.
18. Solve $4^x = 7$ (exact and approximate)
19. Evaluate $\log(0.001)$
20. Condense $\ln(x) - 2\ln(y) + \frac{1}{3}\ln(z)$
21. Solve $3 \cdot 2^x = 48$
22. If $\log_b(5) = 1.465$, find $b$.
23. True/False: $\log(a \cdot b) = \log(a) \cdot \log(b)$
24. How long for $\$1$ to become $\$1{,}000{,}000$ at 10% compounded annually?
25. Evaluate $\log_2(\sqrt{32})$
Show Answer Key

1. $3$  ($4^3=64$)

2. $x = 7$  ($2^7=128$)

3. $2\log(x) + \log(y) - 3\log(z)$

4. $-3$

5. $x = 104$

6. $\log(a^3 \sqrt{b})$

7. $x = \ln(100) \approx 4.605$

8. $\frac{2}{3}$  ($27^{2/3} = 9$)

9. $3000(1.05)^6 \approx \$4{,}020.29$

10. $x = \frac{3}{2}$

11. $10^{\log(30)} = 30$

12. $x = 16$

13. $0.5 = 0.96^t \implies t = \frac{\ln 0.5}{\ln 0.96} \approx 17.0$ years

14. $e^{\ln(9)} = 9$

15. $x = 4$  ($x = -25$ is extraneous)

16. $x = 8^{2/3} = 4$

17. $P(t) = 1000 \cdot 3^{t/5}$; $P(15) = 1000 \cdot 27 = 27{,}000$

18. $x = \frac{\ln 7}{\ln 4} \approx 1.404$

19. $-3$

20. $\ln\!\left(\frac{x \sqrt[3]{z}}{y^2}\right)$

21. $2^x = 16 \implies x = 4$

22. $b^{1.465} = 5 \implies b = 5^{1/1.465} \approx 3$

23. False. Product rule: $\log(ab) = \log(a) + \log(b)$, not $\times$.

24. $1.1^t = 10^6 \implies t = \frac{6\ln 10}{\ln 1.1} \approx 145$ years

25. $\log_2(32^{1/2}) = \frac{1}{2}\log_2(32) = \frac{5}{2}$

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