Q: If $f(x) = 3\ln(x)$, what is the average rate of change of $f$ from $x = 1$ to $x = 2$?

$\frac{3\ln(2) - 3\ln(1)}{2 - 1}$

Q: Which of the following is the correct change of base formula for logarithms?

$\log_b(M) = \frac{\log(M)}{\log(b)}$

Question 1

A radioactive substance decays according to $A = A_0 e^{kt}$, where $k < 0$. If the half-life of the substance is 20 years, what is the value of $k$?

Accepted Answer

$k = \frac{\ln(1/2)}{20}$

Question 2

Which of the following is the correct domain for the function $y = \log(x + 5)$?

Accepted Answer

$x > -5$

Question 3

A population of bacteria grows according to $P(t) = 500e^{0.04t}$. How long will it take for the population to double?

Accepted Answer

$t = \frac{\ln(2)}{0.04}$

Question 4

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

Accepted Answer

Product Rule

Question 5

Given $y = a b^x$, which value of $b$ represents exponential decay?

Accepted Answer

$0 < b < 1$

Question 6

If $f(x) = 3\ln(x)$, what is the average rate of change of $f$ from $x = 1$ to $x = 2$?

Accepted Answer

$\frac{3\ln(2) - 3\ln(1)}{2 - 1}$

Question 7

A medication’s concentration in the bloodstream decreases by 15% per hour. If the initial concentration is $C_0$, what is the concentration after 3 hours?

Accepted Answer

$C_0 	imes 0.85^3$

Question 8

Which of the following is the correct change of base formula for logarithms?

Accepted Answer

$\log_b(M) = \frac{\log(M)}{\log(b)}$

Question 9

Suppose $V(t) = 7e^{-0.16t}$ models the voltage decay in a circuit. What is the percent decrease per second?

Accepted Answer

$1 - e^{-0.16}$

Question 10

Which of the following statements about the graph of $y = \log(x)$ is true?

Accepted Answer

Its domain is $x > 0$

Question 11

If $A = Pe^{rt}$ represents continuous compounding, what does $r$ represent?

Accepted Answer

The rate (as a decimal)

Question 12

Which step is NOT part of the standard process for solving exponential equations using logarithms?

Accepted Answer

Multiply both sides by the exponent

Question 13

Given $y = \log(x + 11)$, what is the vertical asymptote of the graph?

Accepted Answer

$x = -11$

Question 14

A chemical reaction follows $C(t) = 100e^{-0.05t}$. How long until only 25% of the initial concentration remains?

Accepted Answer

$t = \frac{\ln(0.25)}{-0.05}$

Question 15

Which of the following is the correct power rule for logarithms?

Accepted Answer

$\log_b(M^n) = n \log_b(M)$

Question 16

If $y = ab^x$ and $b = 1 + r$ for growth, what does $r$ represent?

Accepted Answer

The percent increase per period (as a decimal)

Question 17

Which of the following is true about the range of an exponential function $y = a b^x$?

Accepted Answer

$(0, \infty)$

Question 18

A scientist models the growth of a cell culture with $N(t) = 2000e^{0.03t}$. What is the average rate of change of $N$ from $t = 0$ to $t = 10$?

Accepted Answer

$\frac{2000e^{0.3} - 2000}{10}$

Question 19

Which of the following is NOT a feature of the graph of $y = e^x$?

Accepted Answer

Vertical asymptote at $x = 0$

Exponential and Logarithmic Models: Concepts, Properties, and Applications

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