Natural Logarithms (ln)

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) $1$
Answer: $0$
$\ln 1 = 0$ because $e^0 = 1$ . Any number (except 0) to the power 0 is 1.
2.
Evaluate $\ln(e^7)$ without a calculator.
Answer: 7
$\ln(e^7)$ asks which power of $e$ gives $e^7$ . The answer is the exponent: $\ln(e^7) = 7$ .
3.
Evaluate $\ln(e^{-3}) + \ln(e^{8})$ without a calculator.
Answer: 5
1. What is $\ln(e^{-3})$ ? -3
2. What is $\ln(e^{8})$ ? 8
3. Add the two results. What is the sum? 5
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) $e^x$
Answer: $\log_e x$
$\ln x$ is the logarithm with base $e$ , so $\ln x = \log_e x$ . $\log_{10} x$ is the common logarithm and $e^x$ is the inverse function of $\ln x$ .
5.
Simplify $e^{\ln 12}$ .
Answer: 12
$\ln 12$ is the power of $e$ that gives $12$ . Raising $e$ to exactly that power gives $12$ back: $e^{\ln 12} = 12$ .
6.
Evaluate $4\ln e - 3\ln 1$ .
Answer: 4
1. What is $\ln e$ ? 1
2. What is $\ln 1$ ? 0
3. What is $4 \cdot 1 - 3 \cdot 0$ ? 4
7.
Solve $\ln x = 2$ . Give $x$ rounded to 2 decimal places.
Answer: 7.39
1. Write in exponential form: $x = e^{\square}$ . What number goes in the box? 2
2. Calculate $e^2$ , rounded to 2 decimal places. 7.39
8.
Which of these has no real value?
1. a) $\ln(-4)$
2. b) $\ln(e^{-2})$
3. c) $\ln 1$
4. d) $\ln 0.5$
Answer: $\ln(-4)$
$\ln(-4)$ has no real value because $e^y$ is always positive. The others exist: $\ln 0.5 \approx -0.693$ , $\ln 1 = 0$ and $\ln(e^{-2}) = -2$ . A negative RESULT is fine, only the argument must be positive.
9.
Use a calculator to find $\ln 50$ . Round to 3 decimal places.
Answer: 3.912
$\ln 50 = 3.91202...$ , so $\ln 50 \approx 3.912$ . Check: $e^{3.912} \approx 50.0$ .
10.
Solve $e^x = 15$ . Round to 3 decimal places.
Answer: 2.708
1. Take $\ln$ of both sides: $x = \ln \square$ . What number goes in the box? 15
2. Calculate $\ln 15$ , rounded to 3 decimal places. 2.708
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) $x = \log 9$
Answer: $x = \ln 9$
Take $\ln$ of both sides: $\ln(e^x) = \ln 9$ , so $x = \ln 9 \approx 2.197$ . Dividing by $e$ does not undo a power, and $\log 9$ uses base 10, not $e$ .
12.
Evaluate $\ln \sqrt{e}$ . Write the answer as a fraction or a decimal.
Answer: 1/2
$\sqrt{e} = e^{1/2}$ , and $\ln(e^{1/2}) = \frac{1}{2}$ . So $\ln \sqrt{e} = \frac{1}{2} = 0.5$ .
13.
Solve $4e^x = 36$ . Round to 3 decimal places.
Answer: 2.197
1. Divide both sides by 4. What is $e^x$ equal to? 9
2. Take $\ln$ of both sides. What is $x = \ln 9$ , rounded to 3 decimal places? 2.197
3. Check: what is $4e^{2.197}$ , rounded to the nearest whole number? 36
14.
Evaluate $e^{\ln 5} \cdot \ln(e^4)$ without a calculator.
Answer: 20
1. What is $e^{\ln 5}$ ? 5
2. What is $\ln(e^4)$ ? 4
3. Multiply the two results. What is the product? 20
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) $x = \ln \frac{7}{2}$
Answer: $x = \frac{\ln 7}{2}$
$\ln(e^{2x}) = 2x$ , so $2x = \ln 7$ and $x = \frac{\ln 7}{2} \approx 0.973$ . Note that $\frac{\ln 7}{2}$ is not the same as $\ln \frac{7}{2} \approx 1.253$ .
16.
Solve $e^x = 0.2$ . Round to 3 decimal places.
Answer: -1.609
Take $\ln$ of both sides: $x = \ln 0.2 = -1.60943...$ , so $x \approx -1.609$ . It is negative because $0.2 < 1$ . Check: $e^{-1.609} \approx 0.200$ .
17.
Solve $e^{3x} = 50$ . Round to 3 decimal places.
Answer: 1.304
1. Take $\ln$ of both sides: $3x = \ln 50$ . What is $\ln 50$ , rounded to 4 decimal places? 3.9120
2. Divide by 3. What is $x$ , rounded to 3 decimal places? 1.304
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.
Answer: 30.54
1. Divide $5000 = 2000e^{0.03t}$ by 2000. What is $e^{0.03t}$ equal to? 2.5
2. Take $\ln$ of both sides: $0.03t = \ln 2.5$ . What is $\ln 2.5$ , rounded to 4 decimal places? 0.9163
3. Divide by 0.03. What is $t$ , rounded to 2 decimal places? 30.54