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

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

Q: Which of the following is NOT a property of logarithms?

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

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 $b > 1$?

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 $b > 1$?

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 $0 < b < 1$?

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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