$$pH = -\log_{10}(2.7 \times 10$$^{-8}$$) \approx 7.57$$

Question 1

Which of the following statements is true about the inverse of a function $$f(x)$$?

Accepted Answer

The graph of $$f^{-1}(x$$)$$ is a reflection of f(x$$)$$ across the line y = x$.

Question 2

Find the inverse of the function $$f(x) = \dfrac{2x+5}{x-1}$$ for $$x 
eq 1.$$

Accepted Answer

$$f^{-1}(x$$) = \dfrac{x+5}{x-2}$$

Question 3

Which property of exponents is illustrated by $$a^m \cdot a^n = a$$^{m+n}$$?

Accepted Answer

Product of Powers Property

Question 4

Simplify $$\left(\dfrac{1}{25}\$$right)^{3/2}$$.

Accepted Answer

$$\dfrac{1}{125}$$

Question 5

Which of the following describes the behavior of the exponential function $$f(x) = a^x$$ when $$a \u003e 1$$?

Accepted Answer

Exponential growth; the function increases as $$x$$ increases.

Question 6

Given $$y = \log_a x$$, which of the following is equivalent in exponential form?

Accepted Answer

$$a^y = x$$

Question 7

Expand $$\log_a(x^5$ y^8$)$$ using logarithm properties.

Accepted Answer

$$5 \log_a x + 8 \log_a y$$

Question 8

If $$\log_2$ x + \log_2$ (x-6) = 2$$, what is the value of $$x$$?

Accepted Answer

$$x = 8$$

Question 9

Which of the following is the correct formula for continuously compounded interest?

Accepted Answer

$$A = Pe$$^{rt}$$

Question 10

A radioactive substance decays according to $$y = y_0$ e$$^{-0.06t}$$. What is the half-life of the substance, to the nearest year?

Accepted Answer

$$t = \dfrac{\ln 2}{0.06} \approx 12$$

Question 11

If the pH of a solution is $$7.68$$, what is the concentration of hydronium ion $$[H_3O^+]$$?

Accepted Answer

$$[H_3O^+] = 10$$^{-7.68}$$

Question 12

Find the pH of a solution with $$[H_3O^+] = 2.7 	imes 10$$^{-8}$$.

Accepted Answer

$$pH = -\log_{10}(2.7 	imes 10$$^{-8}$$) \approx 7.57$$

Question 13

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 14

Which of the following is the domain of the function $$f(x) = \log_2$(x-1) + 5$$?

Accepted Answer

$$x \u003e 1$$

Question 15

If $$f(x) = e$$^{x+1}$$ + 5$$, what is $$f^{-1}(x$$)$$?

Accepted Answer

$$f^{-1}(x$$) = \ln(x-5) - 1$$

Question 16

Which of the following is NOT a property of logarithms?

Accepted Answer

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

Question 17

Which of the following is the range of the exponential function $$f(x) = 2^x$$?

Accepted Answer

$$(0, \infty)$$

Question 18

Which equation best models exponential decay in a real-world scenario?

Accepted Answer

$$y = y_0$ e$$^{-kt}$$, where k$$ \u003e 0$$

Question 19

If $$a \u003e 0$$ and $$a 
eq 1$$, which of the following is true for $$y = \log_a x$$?

Accepted Answer

$$a^y = x$$

Question 20

Which of the following is the correct property for the power of a product?

Accepted Answer

$$(ab)^n = a^n b^n$$

Precalculus Review: Exponential and Logarithmic Functions, Inverses, and Properties of Exponents

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