Question 1

A biologist is studying the growth of a bacterial culture. The population doubles every 3 hours. If the initial population is 500 bacteria, which equation models the population $$P$$ after $$t$$ hours?

Accepted Answer

$$P = 500 	imes 2$$^{t/3}$$

Question 2

Solve for $$x$$: $$\log 5x = \log (2x + 9).$$

Accepted Answer

$$x = 3$$

Question 3

A medication decays in the bloodstream according to $$A = A_0$ e$$^{-0.2t}$$, where A_0$ is the initial amount and t$ is in hours. How long will it take for the medication to decrease to 25% of its original amount?

Accepted Answer

$$t = \frac{\ln 0.25}{-0.2}$$

Question 4

Which of the following is equivalent to $$\log_2$ 8 + \log_2$ 4$$?

Accepted Answer

$$\log_2$ 32$$

Question 5

Solve for $$x$$: $$\log (10 - 4x) = \log (10 - 3x).$$

Accepted Answer

$$x = 0$$

Question 6

A radioactive substance decays according to $$N = N_0$ e$$^{-kt}$$. If the half-life is 5 years, what is the value of k$?

Accepted Answer

$$k = \frac{\ln 2}{5}$$

Question 7

Solve for $$x$$: $$\log (4p - 2) = \log (5p + 5).$$

Accepted Answer

$$p = 7$$

Question 8

If $$\log_a b = 2$$ and $$\log_a c = 3$$, what is $$\log_a (b^2$ c^3$)$$?

Accepted Answer

$$13$$

Question 9

Solve for $$x$$: $$2 \log_x 7 = x.$$

Accepted Answer

$$x = \sqrt{7}$$

Question 10

A sound has an intensity of $$I = 10$$^{-6}$$ W/m^2$. The decibel level is given by L = 10$$ \log_{10} \left( \frac{I}{$$10^{-12}$$} \right)$$. What is the decibel level?

Accepted Answer

$$60$$

Question 11

Solve for $$x$$: $$\log x + \log 8 = 2.$$

Accepted Answer

$$x = \frac{25}{2}$$

Question 12

Which of the following is equivalent to $$\ln (x^2$ - 1) - \ln (x - 1)$$?

Accepted Answer

$$\ln (x + 1)$$

Question 13

Solve for $$x$$: $$\log_2$ (x + 1) - \log_2$ x = \log_2$ 29.$$

Accepted Answer

$$x = 1/28$$

Question 14

A city’s population grows according to $$P = 10000 e$$^{0.03t}$$, where t$ is in years. What is the population after 10 years, rounded to the nearest integer?

Accepted Answer

$$13499$$

Question 15

Solve for $$x$$: $$\ln (x - 3) - \ln (x - 5) = \ln 5.$$

Accepted Answer

$$x = \frac{5}{4}$$

Question 16

If $$\log_2$ x = 5$$, what is the value of $$x$$?

Accepted Answer

$$32$$

Question 17

Solve for $$x$$: $$\log (4x - 5) = \log (2x - 1).$$

Accepted Answer

$$x = 2$$

Question 18

A pH of a solution is defined as $$pH = -\log_{10} [H^+]. If$$ $$[H^+] = 1 	imes 10$$^{-5}$$, what is the pH?

Accepted Answer

$$5$$

Question 19

Solve for $$x$$: $$\log (2a + 9) = \log (7 - 4a).$$

Accepted Answer

$$a = -1$$

Question 20

If $$\log_3$ x = 2$$, what is $$\log_3$ (x^2$)$$?

Accepted Answer

$$4$$

Solving Logarithmic and Exponential Equations: Precalculus Study Guide

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