Question 1

Let $$f(x) = \dfrac{x^2$ - 9}{x^2$ - 4x + 3}. $$Which of the following statements is TRUE about the graph of $$f(x)$$?

Accepted Answer

There is a vertical asymptote at $$x = 3$$ and a hole at $$x = -3.$$

Question 2

Which of the following is the horizontal asymptote of $$f(x) = \dfrac{4x^2$ + 1}{2x^2$ - 5}$$?

Accepted Answer

$$y = 2$$

Question 3

A certain radioactive substance decays according to $$A(t) = 100e$$^{-0.03t}$$, where t$ is in years. How many years will it take for the substance to decay to 50 grams?

Accepted Answer

$$t = \dfrac{\ln(2)}{0.03}$$

Question 4

Which of the following is the correct expansion of $$\log_2$(8x^3$$)$?

Accepted Answer

$$3\log_2$(x) + \log_2$(8)$$

Question 5

Solve for $$x$$: $$\log_3$(x) + \log_3$(x-2) = 2.$$

Accepted Answer

$$x = 2 + \sqrt{13}$$

Question 6

Which of the following is the correct domain of $$f(x) = \ln(x^2$ - 16)$$?

Accepted Answer

$$(-\infty, -4) \cup (4, \infty)$$

Question 7

Rewrite $$\log_7$(20)$$ using natural logarithms.

Accepted Answer

$$\dfrac{\ln(20)}{\ln(7)}$$

Question 8

A $$1000$$ dollar investment grows according to $$A = 1000\left(1 + \dfrac{0.06}{4}\$$right)^{4t}$$. How long will it take for the investment to reach 2000$ dollars?

Accepted Answer

$$t = \dfrac{\ln(2)}{4\ln(1.015)}$$

Question 9

Which of the following is the correct condensed form of $$2\ln(x) - \dfrac{1}{2}\ln(y) + 3\ln(z)$$?

Accepted Answer

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

Question 10

Solve for $$x$$: $$5^{2x-1} = 7$.

Accepted Answer

$$x = \dfrac{\ln(7)}{2\ln(5)} + \dfrac{1}{2}$$

Question 11

Which of the following best describes the graph of $$f(x) = 2\left(\dfrac{1}{4}\right)^x$$?

Accepted Answer

Exponential decay, $$y-$$intercept $$2$$, horizontal asymptote $$y = 0$$

Question 12

Solve the inequality $$\ln(x-1) \leq 0.$$

Accepted Answer

$$1 \u003c x \leq 2$$

Question 13

Which of the following is the $$x-$$intercept of $$f(x) = \dfrac{x-5}{x+2}$$?

Accepted Answer

$$(5, 0)$$

Question 14

Rewrite $$\ln(7) = x in $$exponential form.

Accepted Answer

$$e^x = 7$$

Question 15

Which of the following is the correct domain of $$f(x) = \log(x-3)$$?

Accepted Answer

$$(3, \infty)$$

Question 16

Solve for $$x$$: $$2^x \u003e 16.$$

Accepted Answer

$$x \u003e 4$$

Question 17

Which of the following is the correct condensed form of $$\log_5$(a) + 2\log_5$(b) - \dfrac{1}{2}\log_5$(c)$$?

Accepted Answer

$$\log_5$\left(\dfrac{a b^2$}{\sqrt{c}}\right)$$

Question 18

A population of bacteria triples every 5 hours. Which equation models the population $$P$$ after $$t$$ hours, if the initial population is $$P_0$$?

Accepted Answer

$$P = P_0$ \cdot 3$$^{t/5}$$

Question 19

Solve for $$x$$: $$\log_2$(x-3) = 4.$$

Accepted Answer

$$x = 19$$

Rational, Exponential, and Logarithmic Functions: Study Guide for Exam 4 (Math 225)

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