Real-Life Applications of Logarithmic Functions

1. A scientist is studying the growth of bacteria in a lab. The number of bacteria doubles every hour. If the initial count is 100, how many hours will it take for the bacteria to reach 12,800? Use logarithms to find your answer.

• 5 hours

• 12 hours

• 7 hours

• 10 hours

2. A sound engineer measures the intensity of sound in decibels (dB). The formula for sound intensity is given by I = 10 log10(P/P0), where P is the power of the sound and P0 is a reference power. If the sound intensity is 50 dB, what is the power of the sound if P0 is 1 mW?

• 10 W

• 100 W

• 1000 mW

• 1 W

3. A bank offers an investment that grows according to the formula A = P(1 + r)^t, where A is the amount of money accumulated after n years, P is the principal amount, r is the rate of interest, and t is the time in years. If you want to find out how long it will take for your investment to double, express this in terms of logarithms.

• t = log(1) / log(1 + r)

• t = log(3) / log(1 + r)

• t = log(2) / log(1 + r)

• t = log(2) / log(r)

4. The pH level of a solution is calculated using the formula pH = -log10[H+], where [H+] is the concentration of hydrogen ions. If a solution has a pH of 3, what is the concentration of hydrogen ions?

• 0.0001 M

• 0.1 M

• 0.01 M

• 0.001 M

5. A population of a certain species of fish in a lake is modeled by the function P(t) = P0 * e^(kt), where P0 is the initial population, k is a constant, and t is time in years. If the population is expected to triple in 5 years, how can you express this using logarithms to find the value of k?

• k = ln(3)/3

• k = ln(5)/5

• k = ln(2)/5

• k = ln(3)/5

6. An earthquake's magnitude is measured on a logarithmic scale. If an earthquake has a magnitude of 6.0, how many times more intense is it compared to an earthquake with a magnitude of 4.0? Use the properties of logarithms to solve this problem.

• 200

• 100

• 50

• 10

7. A car's value depreciates according to the formula V(t) = V0 * e^(-rt), where V0 is the initial value, r is the rate of depreciation, and t is time in years. If a car is worth $20,000 now and will be worth $10,000 in 5 years, how can you use logarithms to find the rate of depreciation?

• 0.2000

• 0.0500

• 0.0754

• 0.1386

8. The Richter scale measures the magnitude of earthquakes logarithmically. If an earthquake measures 7.0 on the Richter scale, how much more energy does it release compared to an earthquake measuring 5.0?

• Approximately 100 times more energy

• Approximately 500 times more energy

• Approximately 2000 times more energy

• Approximately 1000 times more energy

9. A company’s profit is modeled by the equation P = 1000 log10(x), where x is the number of units sold. If the company wants to achieve a profit of $5000, how many units must they sell?

• 200000

• 50000

• 75000

• 100000

10. The half-life of a radioactive substance is the time it takes for half of the substance to decay. If the half-life is 10 years, how can you express the remaining amount of substance after 30 years using logarithms?

• 1/4

• 1/8

• 1/2

• 1/16