Question 1

Given the function $$f(x) = 3 \log_{4}(x) + 8$$, what is the derivative $$f'(x)$$?

Accepted Answer

$$f'(x) = \dfrac{3}{x \ln 4}$$

Question 2

Solve for $$x$$: $$4 \cdot 2$$^{2x}$$ - 10 = 22$$

Accepted Answer

$$x = 2$$

Question 3

Which of the following is equivalent to $$\log_{a}\left(\dfrac{b^2$ c^3$}{a^4$}\right)$$ for positive constants $$a, b, c$$?

Accepted Answer

$$2\log$$_{a}$$b + 3\log$$_{a}$$c - 4$$

Question 4

Given $$\ln(x+10) + \ln 5 \leq \ln(2x+17)$$, which of the following describes the solution set for $$x$$?

Accepted Answer

$$x \geq -10$$ and $$x \leq 1.4$$

Question 5

Suppose $$f(x) = \log_{3}(x)$$ and $$g(x) = \log_{3}(81x). $$Which of the following correctly rewrites $$g(x)$$ using properties of logarithms, and what transformation does this represent from $$f(x) to$$ $$g(x)$$?

Accepted Answer

$$g(x) = 4 + \log_{3}(x)$$; a vertical shift up by 4

Question 6

Given the function $$f(x) = \log(3-x) + 2$$, what is the domain of $$f$$?

Accepted Answer

$$x \u003c 3$$

Question 7

Given $$g(x) = \log_{2}(x^3$) + \log_{2}(x)$$, rewrite $$g(x) as a $$single logarithm with no negative exponents.

Accepted Answer

$$g(x) = \log_{2}(x^4$)$$

Question 8

Rewrite $$h(x) = \dfrac{2$$^{x+1}$$}{3$$^{x-2}$$} as a $$single base with an exponent.

Accepted Answer

$$h(x) = \left(\dfrac{2}{3}\right)^x \cdot 2 \cdot 3^2$

Question 9

Solve for $$x in $$the equation $$k(x) = 5 + 6\log_{2}(x) = 17.$$

Accepted Answer

$$x = 64$$

Question 10

For $$m(x) = 2\ln(x) - \ln(7x-10)$$, find all $$x in $$the domain such that $$m(x) = 0.$$

Accepted Answer

$$x = 10$$

Question 11

Given the table: $$x = 2, 4, 8, 16, 32$$ and $$f(x) = 8, 16, 24, 32, 40$$, which model best fits the data?

Accepted Answer

Linear

Question 12

Given $$g(x) = 8.25 - 2\log_{2}(x)$$, what is $$\displaystyle \lim_{x 	o 0^+} g(x)$$?

Accepted Answer

$$+\infty$$

Question 13

If $$f(x) = \log(3-x) + 2$$, what is the vertical asymptote of $$f$$?

Accepted Answer

$$x = 3$$

Question 14

Which of the following best describes the end behavior of $$g(x) = 8.25 - 2\log_{2}(x) as$$ $$x$$ increases without bound?

Accepted Answer

$$g(x) 	o -\infty$$

Question 15

Given $$f(x) = \log_{2}(x)$$, what is the range of $$f$$?

Accepted Answer

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

Question 16

Suppose $$\ln(x+10) + \ln 5 \leq \ln(2x+17). $$Which property of logarithms is most useful for combining the left side?

Accepted Answer

Product property: $$\ln a + \ln b = \ln(ab)$$

Question 17

A rookie card's value decreases after each game played, and the data fits a logarithmic regression. Which equation form best models the value $$V$$ after $$x$$ games?

Accepted Answer

$$V = a + b \ln(x)$$

Question 18

Given $$f(x) = \log_{2}(x)$$ and $$g(x) = \log_{2}(81x)$$, what is the value of $$g(2)$$?

Accepted Answer

$$g(2) = 5$$

Question 19

If $$f(x) = 3\log_{4}(x) + 8$$, what is the range of $$f$$?

Accepted Answer

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

Exponential and Logarithmic Functions Review – Step-by-Step Study Guidance

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