Question 1

Which of the following best describes the domain of the rational function $$h(x) = \frac{x-1}{x-2}$$?

Accepted Answer

All real numbers except $$x = 2$$

Question 2

Suppose you invest $$1000 in a bank account with an annual interest rate of 5\% compounded quarterly. What is the amount in the account after 2 years? Use the formula P = C$$ \left(1 + \frac{r}{n}\$$right)^{nt}$.

Accepted Answer

$$1104.94$$

Question 3

Which of the following statements about the exponential function $$f(x) = b^x is $$TRUE for $$b \u003e 1$$?

Accepted Answer

$$f(x)$$ increases as $$x$$ increases

Question 4

If $$P(t) = 500e$$^{0.03t}$$ models the population of a bacteria colony, what is the population at t = 10$?

Accepted Answer

$$500e^{0.3}$$ \approx 674$$

Question 5

Which of the following is the correct inverse of the function $$f(x) = e^x$$?

Accepted Answer

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

Question 6

A radioactive substance decays according to $$P(t) = 100e$$^{-0.02t}$$. What is the half-life of the substance?

Accepted Answer

$$t = \frac{\ln(2)}{0.02} \approx 34.66$$

Question 7

Which property of logarithms allows you to write $$\log_b(MN) = \log_b(M) + \log_b(N)$$?

Accepted Answer

Product Rule

Question 8

If $$\log_3$(x) = 4$$, what is the value of $$x$$?

Accepted Answer

$$x = 81$$

Question 9

Which of the following best describes the range of the logarithmic function $$f(x) = \log_b(x)$$ for $$b \u003e 1$$?

Accepted Answer

All real numbers

Question 10

A certain medicine decreases in the bloodstream according to $$C(t) = 50e$$^{-0.1t}$$, where C$ is the concentration in mg/L and t$ is in hours. How long until the concentration drops to 10$ mg/L?

Accepted Answer

$$t = \frac{\ln(0.2)}{-0.1} \approx 16.09$$

Question 11

If $$f(x) = 2^x$$ and $$g(x) = \log_2$(x)$$, which of the following is TRUE?

Accepted Answer

$$f$$ and $$g$$ are inverse functions

Question 12

Which of the following expressions is equivalent to $$\ln(e$$^{5x}$$)$$?

Accepted Answer

$$5x$$

Question 13

A population doubles every 8 years. What is the exponential growth rate $$k in $$the model $$P(t) = Ce$$^{kt}$$?

Accepted Answer

$$k = \frac{\ln(2)}{8}$$

Question 14

Which of the following is TRUE about the function $$f(x) = b^x$$ for $$0 \u003c b \u003c 1$$?

Accepted Answer

$$f(x)$$ decreases as $$x$$ increases

Question 15

If $$P(t) = 200e$$^{kt}$$ and P(5) = 400$, what is the value of k$?

Accepted Answer

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

Question 16

Which of the following is NOT a property of logarithms?

Accepted Answer

$$\log_b(M+N) = \log_b(M) + \log_b(N)$$

Question 17

If $$f(x) = \log_5$(x)$$, what is the value of $$f(1/25)$$?

Accepted Answer

$$-2$$

Question 18

A sample of a chemical has a half-life of 3 hours. If the initial amount is 80 grams, how much remains after 9 hours? Use $$P(t) = Ce$$^{kt}$$ and k$$ = \frac{\ln(1/2)}{3}$$.

Accepted Answer

$$10$$ grams

Question 19

Which of the following correctly expresses the solution for $$x in $$the equation $$3^x = 20$$?

Accepted Answer

$$x = \frac{\ln(20)}{\ln(3)}$$

Exponential and Logarithmic Functions: Definitions, Properties, and Applications

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