Logarithm Word Problems

Show your work for each problem.

1.
An earthquake of magnitude $6$ is compared with one of magnitude $4$ on the Richter scale. How many times larger is the amplitude of the stronger earthquake?
1. a)20
2. b)1000
3. c)2
4. d)100
Answer: 100
The ratio is $10^{6 - 4} = 10^2 = 100$ . A difference of $2$ means two factors of $10$ , not "2 times".
2.
A solution has a hydrogen ion concentration of $[H^+] = 10^{-5}$ mol/L. What is its pH?
1. a) $0.00001$
2. b) $10^5$
3. c) $-5$
4. d) $5$
Answer: $5$
$\text{pH} = -\log(10^{-5}) = -(-5) = 5$ . The minus sign in the formula makes the pH positive.
3.
You invest 1000 dollars at $5\%$ interest compounded once a year. Which expression gives the time $t$ , in years, until the money doubles?
1. a) $\frac{\ln 2000}{\ln 1000}$
2. b) $\frac{\ln 2}{\ln 0.05}$
3. c) $\frac{\ln 2}{\ln 1.05}$
4. d) $2 \cdot \ln 1.05$
Answer: $\frac{\ln 2}{\ln 1.05}$
$1000 \cdot 1.05^t = 2000$ gives $1.05^t = 2$ , so $t \ln 1.05 = \ln 2$ and $t = \frac{\ln 2}{\ln 1.05} \approx 14.21$ . The growth factor is $1.05$ , not the rate $0.05$ .
4.
Sound A has a level of $90$ dB and sound B a level of $60$ dB. How many times as intense is sound A as sound B?
1. a)300
2. b)1000
3. c)3
4. d)30
Answer: 1000
The difference is $30$ dB, so the ratio is $10^{30/10} = 10^3 = 1000$ .
5.
A substance loses $8\%$ of its mass every year. Which equation gives the number of years $t$ until only half of it remains?
1. a) $t = \frac{\ln 0.5}{\ln 1.08}$
2. b) $t = \frac{\ln 0.5}{\ln 0.08}$
3. c) $t = 0.5 \cdot \ln 0.92$
4. d) $t = \frac{\ln 0.5}{\ln 0.92}$
Answer: $t = \frac{\ln 0.5}{\ln 0.92}$
Each year $92\%$ remains, so the decay factor is $0.92$ : $0.92^t = 0.5$ , $t \ln 0.92 = \ln 0.5$ and $t = \frac{\ln 0.5}{\ln 0.92} \approx 8.31$ years.
6.
A solution has $[H^+] = 0.001$ mol/L. What is its pH? (The answer is a whole number.)
Answer: 3
$0.001 = 10^{-3}$ , so $\text{pH} = -\log(10^{-3}) = 3$ .
7.
A sound has intensity $I = 10^7 \cdot I_0$ . What is its level $L = 10\log\left(\frac{I}{I_0}\right)$ in decibels? (The answer is a whole number.)
Answer: 70
$L = 10 \log(10^7) = 10 \times 7 = 70$ dB.
8.
Find the pH of a solution with $[H^+] = 2.5 \times 10^{-6}$ mol/L. Round to two decimal places.
Answer: 5.60
$\log(2.5 \times 10^{-6}) = \log 2.5 - 6 \approx 0.39794 - 6 = -5.60206$ , so $\text{pH} \approx 5.60$ .
9.
An earthquake of magnitude $7.4$ is compared with one of magnitude $5.6$ . How many times larger is the amplitude of the stronger one? Round to the nearest whole number.
Answer: 63
$10^{7.4 - 5.6} = 10^{1.8} \approx 63.096$ , which rounds to $63$ .
10.
A town has $25\,000$ inhabitants and grows by $3\%$ per year. After how many years will it reach $40\,000$ inhabitants? Round to one decimal place.
Answer: 15.9
$1.03^t = \frac{40\,000}{25\,000} = 1.6$ , so $t = \frac{\ln 1.6}{\ln 1.03} \approx \frac{0.470004}{0.029559} \approx 15.90$ , which rounds to $15.9$ years.
11.
Lemon juice has a hydrogen ion concentration of $[H^+] = 0.01$ mol/L. Find its pH.
Answer: 2
1. Write $0.01$ as a power of 10: $0.01 = 10^{n}$ . What is $n$ ? -2
2. What is $\log(0.01)$ ? -2
3. What is the pH, $-\log(0.01)$ ? 2
12.
A hair dryer produces a sound $1\,000\,000$ times as intense as the threshold of hearing. Find its level in decibels.
Answer: 60
1. Write $1\,000\,000 = 10^{n}$ . What is $n$ ? 6
2. What is $\log(1\,000\,000)$ ? 6
3. What is $L = 10 \log(1\,000\,000)$ in dB? 60
13.
A cup of coffee has a pH of $5$ . Find its hydrogen ion concentration $[H^+]$ in mol/L, written as a decimal number.
Answer: 0.00001
1. From $-\log[H^+] = 5$ , what is $\log[H^+]$ ? -5
2. So $[H^+] = 10^{k}$ . What is $k$ ? -5
3. Write $10^{-5}$ as a decimal number. 0.00001
14.
Heavy traffic measures $85$ dB and a normal conversation $60$ dB. How many times as intense is the traffic noise? Round to the nearest whole number.
Answer: 316
1. What is the difference in decibels? 25
2. The intensity ratio is $10^{x}$ with $x = \frac{25}{10}$ . What is $x$ ? 2.5
3. Evaluate $10^{2.5}$ and round to the nearest whole number. 316
15.
Money is invested at $6\%$ interest compounded once a year. How many full years does it take until the balance has at least doubled?
Answer: 12
1. By which growth factor is the balance multiplied each year? 1.06
2. Compute $t = \frac{\ln 2}{\ln 1.06}$ , rounded to two decimal places. 11.90
3. Interest is paid once a year. How many full years are needed? 12
16.
A lab sample contains $200$ mg of a substance with a half-life of $5$ hours. After how many hours will only $20$ mg remain? Round to one decimal place.
Answer: 16.6
1. In $200 \cdot 0.5^{t/5} = 20$ , what is $0.5^{t/5} = \frac{20}{200}$ ? 0.1
2. Compute $\frac{t}{5} = \frac{\ln 0.1}{\ln 0.5}$ , rounded to two decimal places. 3.32
3. Multiply by 5. What is $t$ in hours, rounded to one decimal place? 16.6
17.
You invest 3000 dollars at $4\%$ interest compounded continuously, so $A = 3000e^{0.04t}$ . After how many years does the balance reach 4500 dollars? Round to two decimal places.
Answer: 10.14
1. Divide both sides by 3000: $e^{0.04t} = \frac{4500}{3000}$ . What is the right side? 1.5
2. Take ln: $0.04t = \ln 1.5$ . What is $\ln 1.5$ , rounded to four decimal places? 0.4055
3. Divide by $0.04$ . What is $t$ , rounded to two decimal places? 10.14
18.
A car bought for $24\,000$ dollars loses $15\%$ of its value every year. After how many full years is it first worth less than $6000$ dollars?
Answer: 9
1. What is $\frac{6000}{24\,000}$ ? 0.25
2. By which factor is the value multiplied each year? 0.85
3. Solve $0.85^t = 0.25$ : compute $t = \frac{\ln 0.25}{\ln 0.85}$ , rounded to two decimal places. 8.53
4. The value drops once a year. After how many full years is it first below the target? 9