Question 1

Which of the following statements about the function $$y = \log_a(x) is $$TRUE?

Accepted Answer

It passes through the point $$(1, 0)$$ for any base $$a \u003e 0, a 
eq 1.$$

Question 2

If $$y = \log_3$(x)$$, which equation expresses $$x in $$terms of $$y$$?

Accepted Answer

$$x = 3^y$$

Question 3

Which of the following is equivalent to $$\log_a\left(\dfrac{x}{y}\right)$$?

Accepted Answer

$$\log_a(x) - \log_a(y)$$

Question 4

Simplify $$\ln(e$$^{2x}$$) as $$much as possible.

Accepted Answer

$$2x$$

Question 5

Which of the following is the correct change of base formula for $$\log_5$(x) in $$terms of natural logarithms?

Accepted Answer

$$\log_5$(x) = \dfrac{\ln(x)}{\ln(5)}$$

Question 6

Solve for $$x$$: $$e^{\ln(2)x + 3} = 8$.

Accepted Answer

$$x = \dfrac{\ln(8) - 3}{\ln(2)}$$

Question 7

Which of the following expressions is equivalent to $$\ln\left(\dfrac{1}{x}\right)$$?

Accepted Answer

$$-\ln(x)$$

Question 8

If $$\ln(y + 2) + \ln(x) = 4x$$, solve for $$y in $$terms of $$x.$$

Accepted Answer

$$y = \dfrac{e$$^{4x}$$}{x} - 2$$

Question 9

Simplify $$5^{\log_5(x^2)} as $$much as possible.

Accepted Answer

$$x^2$$

Question 10

Which of the following is a property of the graph of $$y = \ln(x)$$?

Accepted Answer

It has a vertical asymptote at $$x = 0.$$

Question 11

Which of the following is equivalent to $$\ln(x^y)$$?

Accepted Answer

$$y \ln(x)$$

Question 12

A scientist is analyzing the pH of a solution, defined as $$\mathrm{pH} = -\log_{10}([H^+]). If$$ $$[H^+] = 10$$^{-4}$$, what is the pH?

Accepted Answer

$$4$$

Question 13

Simplify $$\log_4$(16x) as $$much as possible.

Accepted Answer

$$2 + \log_4$(x)$$

Question 14

Which of the following is equivalent to $$\log_2$(8)$$?

Accepted Answer

$$3$$

Question 15

A radioactive substance decays according to $$N(t) = N_0$ e$$^{-kt}$$. If after 5$ years, N(5$$) = \dfrac{$$N_0$$}{2}$$, what is the value of k$?

Accepted Answer

$$k = \dfrac{\ln(2)}{5}$$

Question 16

Simplify $$\log_8$(e^{\ln(2) 	an x}$$)$ as much as possible.

Accepted Answer

$$\dfrac{1}{3} 	an x$$

Question 17

If $$\ln(x) = 2$$, what is the value of $$x$$?

Accepted Answer

$$e^2$$

Question 18

Which of the following is the correct simplification for $$4^{\log_{16}(x$$)}$$?

Accepted Answer

$$x^{1/2}$$

Question 19

If $$\log_a(x) = y$$, which of the following is always true?

Accepted Answer

$$a^y = x$$

Question 20

Which of the following is NOT a property of logarithms?

Accepted Answer

$$\log_a(x + y) = \log_a(x) + \log_a(y)$$

Logarithms and Their Properties: Pre-Calculus Review for Calculus

Study Guide - Practice Questions