Logarithmic Applications: Growth, Decay, and Sound Intensity

A scientist is studying the growth of bacteria. 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 a logarithmic equation to find the answer.

• 7 hours

• 5 hours

• 10 hours

• 12 hours

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?

• 1000 mW

• 1500 mW

• 500 mW

• 2000 mW

A city’s population is modeled by the equation P(t) = P0 * e^(kt), where P0 is the initial population, k is a constant, and t is time in years. If the population doubles in 5 years, what is the value of k? Use logarithmic properties to solve.

• ln(2) / 5

• ln(4) / 5

• ln(2) / 10

• ln(3) / 5

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 compared to an earthquake that measures 4.0?

• 200

• 10

• 50

• 100

A car's speed is increasing exponentially. If the speed of the car is modeled by the equation v(t) = v0 * e^(rt), where v0 is the initial speed, r is the rate of increase, and t is time in seconds. If the car reaches a speed of 80 km/h in 10 seconds, what is the value of r?

• 0.0323

• 0.1285

• 0.0456

• 0.0647

A bank offers an investment that grows according to the formula A = P(1 + r)^t. If you want to find out how long it will take for your investment to double, what logarithmic property can you use to solve for t?

• Utilize the property of logarithms: log(a^b) = b/a to determine t.

• Apply the property of logarithms: log(a/b) = log(a) - log(b) to find t.

• Use the property of logarithms: log(a + b) = log(a) + log(b) to solve for t.

• Use the property of logarithms: log(a^b) = b*log(a) to solve for t.

The pH scale is logarithmic, where pH = -log10[H+]. If a solution has a pH of 3, what is the concentration of hydrogen ions [H+] in the solution?

• 0.001 M

• 0.01 M

• 0.0001 M

• 0.1 M

A population of fish in a lake is modeled by the equation N(t) = N0 * log(t + 1). If the initial population is 50 and you want to find out when the population reaches 100, what equation will you set up using logarithmic properties?

• 2 = log(t + 1) => t + 1 = 10^3 => t = 999

• 2 = log(t + 2) => t + 2 = 10^2 => t = 98

• 2 = log(t + 1) => t + 1 = 10^2 => t = 99

• 2 = log(t - 1) => t - 1 = 10^2 => t = 101

A smartphone app tracks the number of downloads, which increases logarithmically. If the app had 1,000 downloads in the first month and 10,000 downloads in the second month, how can you express the relationship between time and downloads using a logarithmic function?

• D(t) = 1000 * log(10 * t)

• D(t) = 1000 * log(t)

• D(t) = 1000 * log(5 * t)

• D(t) = 1000 * log(2 * t)

A researcher is analyzing the decay of a radioactive substance. The amount of substance remaining after t years is given by A(t) = A0 * e^(-kt). If the half-life of the substance is 5 years, how can you use logarithmic properties to find the value of k?

• k = 5 / ln(2)

• k = ln(2) / 10

• k = ln(2) / 5

• k = ln(3) / 5