Natural Logarithms (ln)

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Show your work for each problem.

1.
What is the value of $\ln 1$ ?
1. a) $0$
2. b) $e$
3. c) $-1$
4. d) $1$
2.
Evaluate $\ln(e^7)$ without a calculator.
3.
Evaluate $\ln(e^{-3}) + \ln(e^{8})$ without a calculator.
4.
Which expression means the same as $\ln x$ ?
1. a) $\log_2 x$
2. b) $\log_e x$
3. c) $\log_{10} x$
4. d) $e^x$
5.
Simplify $e^{\ln 12}$ .
6.
Evaluate $4\ln e - 3\ln 1$ .
7.
Solve $\ln x = 2$ . Give $x$ rounded to 2 decimal places.
8.
Which of these has no real value?
1. a) $\ln(-4)$
2. b) $\ln(e^{-2})$
3. c) $\ln 1$
4. d) $\ln 0.5$
9.
Use a calculator to find $\ln 50$ . Round to 3 decimal places.
10.
Solve $e^x = 15$ . Round to 3 decimal places.
11.
What is the exact solution of $e^x = 9$ ?
1. a) $x = \ln 9$
2. b) $x = \frac{9}{e}$
3. c) $x = e^9$
4. d) $x = \log 9$
12.
Evaluate $\ln \sqrt{e}$ . Write the answer as a fraction or a decimal.
13.
Solve $4e^x = 36$ . Round to 3 decimal places.
14.
Evaluate $e^{\ln 5} \cdot \ln(e^4)$ without a calculator.
15.
Which value of $x$ solves $e^{2x} = 7$ ?
1. a) $x = 2\ln 7$
2. b) $x = \frac{\ln 7}{2}$
3. c) $x = \frac{7}{2e}$
4. d) $x = \ln \frac{7}{2}$
16.
Solve $e^x = 0.2$ . Round to 3 decimal places.
17.
Solve $e^{3x} = 50$ . Round to 3 decimal places.
18.
A town's population grows as $P = 2000e^{0.03t}$ , where $t$ is in years. After how many years will the population reach 5000? Round to 2 decimal places.