Logarithmic Applications: Evaluating & Solving in Real Life

1. A scientist measures the pH level of a solution, which is given by the formula pH = -log[H+]. If the concentration of hydrogen ions [H+] is 0.001 M, what is the pH of the solution?

• 1

• 3

• 7

• 10

2. The population of a certain bacteria doubles every 3 hours. If the initial population is 500, how long will it take for the population to reach 8000? Use logarithms to solve the exponential equation.

• 12 hours

• 10 hours

• 15 hours

• 6 hours

3. A sound intensity level is measured in decibels (dB) using the formula L = 10 log(I/I0), where I is the intensity of the sound and I0 is the reference intensity (1 x 10^-12 W/m²). If a sound has an intensity of 1 x 10^-10 W/m², what is its sound level in decibels?

• 30 dB

• 20 dB

• 10 dB

• 50 dB

4. The half-life of a radioactive substance is 5 years. If you start with 80 grams, how much will remain after 15 years? Use logarithms to solve the exponential decay problem.

• 20 grams

• 10 grams

• 5 grams

• 40 grams

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 of $1000 to double at an interest rate of 5%, what logarithmic expression will you use?

• t = log(2) / log(0.05)

• t = log(2) / log(2)

• t = log(2) / log(1.05)

• t = log(3) / log(1.05)

6. An earthquake's magnitude is measured on the Richter scale using the formula M = log(I/I0), where I is the intensity of the earthquake and I0 is a reference intensity. If an earthquake has an intensity of 1000 times the reference intensity, what is its magnitude?

• 4

• 3

• 2

• 5

7. A car depreciates in value according to the formula V = V0 e^(-kt), where V0 is the initial value, k is the depreciation constant, and t is time in years. If a car's initial value is $20,000 and it depreciates at a rate of 15% per year, how long will it take for the car's value to drop below $10,000?

• 5.10 years

• 3.25 years

• 6.75 years

• 4.62 years

8. The formula for compound interest is A = P(1 + r/n)^(nt). If you invest $2000 at an annual interest rate of 4% compounded quarterly, how long will it take for your investment to grow to $3000? Use logarithms to find the answer.

• 12.75 years

• 11.18 years

• 9.2 years

• 10.5 years

9. A certain type of bacteria grows according to the model N(t) = N0 e^(rt), where N0 is the initial quantity, r is the growth rate, and t is time. If the initial quantity is 200 and the growth rate is 0.03, how long will it take for the quantity to reach 500?

• 40.10

• 25.50

• 30.66

• 35.75

10. The formula for the time it takes for a population to reach a certain size is given by t = (log(N/N0))/(log(1 + r)), where N is the final population, N0 is the initial population, and r is the growth rate. If a population starts at 100 and grows to 400 with a growth rate of 0.05, how long will it take to reach that size?

• 28.37

• 15.25

• 50.10

• 20.75