Logarithmic Equations: Growth Rates and Conversions

A population of bacteria doubles every 3 hours. If the initial population is 100, write a logarithmic equation to represent the population after t hours. How many hours will it take for the population to reach 800?

• 9 hours

• 15 hours

• 6 hours

• 12 hours

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? Convert this to a logarithmic form.

• 0.01 moles per liter

• 0.00001 moles per liter

• 1 mole per liter

• 0.0001 moles per liter

A car's value decreases over time according to the function 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 after 5 years is $10,000, express this in logarithmic form to find the value of k.

• k = (ln(10000) - ln(V0)) / 5

• k = ln(V0) / 5

• k = (ln(V0) - ln(10000)) / 5

• k = (ln(10000) + ln(V0)) / 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?

• 10

• 50

• 200

• 100

A certain investment 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, convert this equation into logarithmic form and solve for t.

• t = 2 / (1 + r)

• t = (1 + r) / 2

• t = log(1 + r) / log(2)

• t = log(2) / log(1 + r)

The intensity of sound is measured in decibels (dB), which is a logarithmic scale. If a sound is measured at 80 dB, what is the intensity level in watts per square meter?

• 0.001 W/m^2

• 0.1 W/m^2

• 0.01 W/m^2

• 0.0001 W/m^2

A smartphone's battery life decreases logarithmically. If a fully charged battery lasts for 10 hours, how long will it last after 3 hours of use? Write a logarithmic equation to represent this situation.

• Less than 10 hours

• Exactly 10 hours

• 5 hours

• More than 10 hours

The formula for the half-life of a substance is given by t = (ln(2)/k). If the decay constant k is 0.1, calculate the half-life and express your answer in logarithmic form.

• 5 * ln(2)

• ln(2)

• 20 * ln(2)

• 10 * ln(2)

A tree's height can be modeled by the equation H(t) = H0 * log(t + 1), where H0 is the initial height and t is time in years. If the initial height is 2 meters, find the height of the tree after 5 years and express it in logarithmic form.

• 2 * log(5)

• 2 * log(6)

• 3 * log(6)

• 2 + log(6)