Mastering Logarithmic Functions and Models in Real Life

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?

• 3

• 1

• 7

• 10

The population of a certain city is modeled by the function P(t) = 5000 * e^(0.03t), where t is the number of years since 2000. How many years will it take for the population to reach 10,000?

• 30.2

• 15.5

• 23.1

• 18.7

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. If a sound has an intensity of 0.0001 W/m², what is its sound level in decibels?

• 60 dB

• 100 dB

• 40 dB

• 80 dB

The half-life of a radioactive substance is 5 years. If you start with 80 grams, how much will remain after 15 years? Use the formula N(t) = N0 * (1/2)^(t/T) where N0 is the initial amount, t is time, and T is the half-life.

• 5 grams

• 40 grams

• 20 grams

• 10 grams

A bank offers an investment that grows according to the formula 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?

• 25 years

• 17.67 years

• 10 years

• 12.5 years

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 an earthquake that measures 4.0?

• 200

• 10

• 100

• 50

A car's value depreciates according to the model V(t) = V0 * e^(-kt), where V0 is the initial value, k is the depreciation constant, and t is time in years. If a car is initially worth $20,000 and k = 0.15, what will its value be after 10 years?

• 3000

• 8000

• 4462.60

• 15000

The time it takes for a computer to process a certain task is modeled by T(n) = log(n) + 2, where n is the number of tasks. If a computer can process 100 tasks, how long will it take?

• 8

• 6

• 2

• 4

A bacteria culture grows according to the model N(t) = N0 * 2^(t/3), where N0 is the initial number of bacteria and t is time in hours. If you start with 50 bacteria, how many will there be after 9 hours?

• 800

• 400

• 600

• 200

A light source has an intensity of 1000 lumens. If the intensity decreases logarithmically with distance, how much intensity will be felt at a distance of 10 meters? Use the formula I = I0 / (d^2) where I0 is the initial intensity and d is the distance.

• 100 lumens

• 10 lumens

• 1 lumen

• 50 lumens