Mastering Logs and Exponents: Real-World Applications

A population of bacteria doubles every 3 hours. If there are initially 500 bacteria, how many will there be after 12 hours? Use logarithms to find the time it takes for the population to reach 8000.

• 4000 bacteria after 12 hours

• 6000 bacteria after 12 hours; it takes 9 hours to reach 8000.

• 8000 bacteria after 12 hours; it takes 12 hours to reach 8000.

• 10000 bacteria after 12 hours; it takes 15 hours to reach 8000.

The pH level of a solution is measured on a logarithmic scale. If a solution has a pH of 4, what is the hydrogen ion concentration in moles per liter?

• 0.00001 moles per liter

• 0.1 moles per liter

• 0.0001 moles per liter

• 0.01 moles per liter

A car depreciates in value according to the formula V = P(0.85)^t, where V is the value after t years and P is the initial price. If the car is worth $5000 after 5 years, what was its original price?

• 8000

• 9500

• 10000

• 11266.67

The half-life of a radioactive substance is 10 years. If you start with 80 grams, how much will remain after 30 years? Use logarithms to solve for the remaining amount.

• 5 grams

• 10 grams

• 20 grams

• 40 grams

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 a reference intensity. If a sound has an intensity of 0.001 W/m², what is its decibel level?

• 100 dB

• 90 dB

• 80 dB

• 70 dB

The formula for compound interest is A = P(1 + r/n)^(nt). If you invest $1000 at an annual interest rate of 5% compounded quarterly, how long will it take for your investment to double? Use logarithms to find the answer.

• 30 years

• 25 years

• 17.67 years

• 10 years

A certain species of fish grows exponentially according to the model P(t) = P0e^(kt). If the population is 200 at time t=0 and grows to 800 in 5 years, what is the growth constant k?

• 0.5

• 0.75

• 0.1

• 0.3218875824868201

The Richter scale measures the magnitude of earthquakes logarithmically. If an earthquake measures 6.0 on the Richter scale, how many times more intense is it than one that measures 4.0?

• 50

• 10

• 200

• 100

A certain investment grows according to the formula A = Pe^(rt). If an investment of $2000 grows to $5000 in 10 years, what is the annual interest rate r? Use logarithms to solve for r.

• 0.0916

• 0.08

• 0.12

• 0.05

The formula for the volume of a gas is given by the ideal gas law PV = nRT. If the pressure is doubled while keeping the temperature constant, what happens to the volume? Use logarithmic properties to explain your answer.

• The volume increases by 50%.

• The volume remains the same.

• The volume is halved.

• The volume is doubled.