Question 1

Which of the following statements correctly describes the relationship between the exponential function $$f(x) = a^x$$ and its inverse?

Accepted Answer

The inverse of $$f(x) = a^x is$$ $$f^{-1}(x$$) = \log_a x$$, and f(f$$^{-1}$$(x)) = x$$ for all $$x in $$the domain of $$f^{-1}.$$

Question 2

Convert the logarithmic equation $$\log_5$ (2x - 1) = 3 to $$its equivalent exponential form.

Accepted Answer

$$2x - 1 = 5^3$

Question 3

Which of the following is the correct domain of the function $$f(x) = \log_a x$$ where $$a \u003e 0$$ and $$a 
eq 1$$?

Accepted Answer

$$(0, +\infty)$$

Question 4

Evaluate $$\log_2$ \left( \dfrac{1}{16} \right)$$ without a calculator.

Accepted Answer

$$-4$$

Question 5

Which property of logarithms allows you to write $$\log_a (xy) as a $$sum?

Accepted Answer

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

Question 6

Solve for $$x$$: $$e^{2x} = 7$.

Accepted Answer

$$x = \dfrac{1}{2} \ln 7$$

Question 7

Which of the following is the correct change of base formula for logarithms?

Accepted Answer

$$\log_a x = \dfrac{\log_b x}{\log_b a}$$

Question 8

If $$\log_3$ x + \log_3$ (x+1) = \log_3$ 12$$, what is the solution for $$x$$?

Accepted Answer

$$x = 3$$

Question 9

Which of the following best describes the range of the function $$f(x) = \log_a x$$?

Accepted Answer

$$(-\infty, +\infty)$$

Question 10

Rewrite the expression $$2 \log_5$ x + \log_5$ y - 3 \log_5$ z as a $$single logarithm.

Accepted Answer

$$\log_5$ \left( \dfrac{x^2$ y}{z^3$} \right)$$

Question 11

A certain radioactive substance decays according to the formula $$A(t) = A_0$ e$$^{-kt}$$. If after 5 years, only 60% of the original amount remains, what is the value of k$? (Round to 3 decimal places.)

Accepted Answer

$$k = 0.102$$

Question 12

Which of the following is NOT a property of logarithms?

Accepted Answer

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

Question 13

Solve for $$x$$: $$4^x = 32.$$

Accepted Answer

$$x = \dfrac{5}{2}$$

Question 14

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

Accepted Answer

$$x = e^2$

Question 15

Which of the following points is on the graph of $$f(x) = \log_2$ x$$?

Accepted Answer

$$(8, 3)$$

Question 16

Given $$\log_{10} 142 = 2.1523$$ and $$\log_{10} 10 = 1$$, what is $$\log_{10} (1420)$$?

Accepted Answer

$$3.1523$$

Question 17

Which of the following best describes the graph of $$f(x) = \log_a x$$ for $$0 \u003c a \u003c 1$$?

Accepted Answer

The graph is decreasing for $$x \u003e 0$$ and has a vertical asymptote at $$x = 0.$$

Question 18

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

Accepted Answer

$$a^y = x$$

Question 19

Solve for $$x$$: $$\ln (x^2$) = 4.$$

Accepted Answer

$$x = e^2$ or x$$ = $$-e^2$$

Logarithmic Functions and Their Properties in Business Calculus

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