Real-Life Applications: Graphing and Interpreting Logarithms

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.

• 7 hours

• 10 hours

• 5 hours

• 12 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 the reference power. If the sound intensity is 30 dB, what is the power of the sound?

• 500 mW

• 2000 mW

• 1500 mW

• 1000 mW

3. A population of a certain species of fish in a lake is modeled by the function P(t) = 500 log(t + 1), where P is the population and t is the time in years. How many years will it take for the fish population to reach 1,500?

• 999

• 250

• 750

• 1200

4. The Richter scale measures the magnitude of earthquakes logarithmically. If an earthquake measures 5.0 on the Richter scale, how many times more intense is it compared to an earthquake that measures 3.0?

• 100

• 50

• 10

• 200

5. 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, how would you use logarithms to solve for t?

• t = log(P) / log(1 + r)

• t = log(A) / log(r)

• t = log(2) / log(1 + r)

• t = log(1) / log(1 + r)

6. A car's value depreciates according to the model V(t) = V0 e^(-kt), where V0 is the initial value, k is a constant, and t is time in years. If the car's value is $10,000 after 5 years, what was its initial value? Use logarithms to find your answer.

• V0 = 5000 e^(5k)

• V0 = 10000 e^(10k)

• V0 = 10000 e^(-5k)

• V0 = 10000 e^(5k)

7. 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 4, what is the concentration of hydrogen ions?

• 0.0001 M

• 0.00001 M

• 0.001 M

• 0.01 M

8. A certain radioactive substance has a half-life of 3 years. If you start with 80 grams, how much will remain after 9 years? Use logarithmic functions to determine your answer.

• 10 grams

• 40 grams

• 20 grams

• 5 grams

9. The formula for the half-life of a substance is given by t = (ln(2)/k), where k is the decay constant. If the decay constant of a substance is 0.1, what is its half-life?

• 6.93

• 9.21

• 4.15

• 12.50

10. A company’s revenue can be modeled by the function R(x) = 200 log(x + 1), where x is the number of years since the company was founded. How much revenue will the company generate after 4 years?

• 200.0

• 120.3

• 139.8

• 150.5