Q: A population grows exponentially at a rate of 15% per year. How long will it take for the population to double?

$t = \frac{\ln 2}{0.15} \approx 4.62$ years

Q: Given the compound interest formula $A = P(1 + r)^t$, how long will it take for $1000$ to grow to $3000$ at $12.5\%$ compounded annually?

$t = \frac{\ln 3}{\ln 1.125} \approx 9.69$ years

Q: A concrete cures at a rate of $6.5\%$ per day. How many days will it take to reach a compressive strength of $20$ MPa if the maximum strength is $30$ MPa?

$t = \frac{\ln(1/3)}{-0.065} \approx 16.9$ days

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

$\log_b a = \frac{\log_k a}{\log_k b}$

Q: If the demand function is $p = 4000 - 10x$ and $x = 40$, what is the total revenue $R(x)$?

$R(40) = (4000 - 10 \times 40) \times 40 = 2400 \times 40 = 96000$

Question 1

Which of the following is the correct solution to the exponential equation $5^{2x-1} = 125$?

Accepted Answer

$x = 2$

Question 2

Given the equation $3^{x^2-2x} = \frac{1}{3}$, what is the value of $x$?

Accepted Answer

$x = 1$

Question 3

Solve for $x$: $e^{x^2-x} = e^6$

Accepted Answer

$x = 3$ or $x = -2$

Question 4

What is the exact solution to $3^{x+2} = 7$?

Accepted Answer

$x = \frac{\log 7}{\log 3} - 2$

Question 5

If $e^x = 3$, what is the value of $x$ to four decimal places?

Accepted Answer

$x = \ln 3 \approx 1.0986$

Question 6

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

Accepted Answer

Product Rule

Question 7

Convert the logarithmic equation $\log_5 25 = 2$ to its equivalent exponential form.

Accepted Answer

$25 = 5^2$

Question 8

What is the domain of the function $f(x) = \ln(4 - x^2)$?

Accepted Answer

$|x| < 2$

Question 9

Solve for $x$: $\log_2(x) + \log_2(x-2) = 3$

Accepted Answer

$x = 4$

Question 10

A population grows exponentially at a rate of 15% per year. How long will it take for the population to double?

Accepted Answer

$t = \frac{\ln 2}{0.15} \approx 4.62$ years

Question 11

If $p = 3000e^{-0.05x}$ is the demand function and the price is $p = 149.40$, how many units $x$ are demanded (to the nearest unit)?

Accepted Answer

$x = 60$

Question 12

Given the compound interest formula $A = P(1 + r)^t$, how long will it take for $1000$ to grow to $3000$ at $12.5\%$ compounded annually?

Accepted Answer

$t = \frac{\ln 3}{\ln 1.125} \approx 9.69$ years

Question 13

Which of the following is the correct application of the Power Rule for logarithms?

Accepted Answer

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

Question 14

If $\log_{10} x = 2$, what is the value of $x$?

Accepted Answer

$x = 100$

Question 15

Solve for $x$: $\ln(x+2) + \ln x = \ln(x+12)$

Accepted Answer

$x = 3$

Question 16

A concrete cures at a rate of $6.5\%$ per day. How many days will it take to reach a compressive strength of $20$ MPa if the maximum strength is $30$ MPa?

Accepted Answer

$t = \frac{\ln(1/3)}{-0.065} \approx 16.9$ days

Question 17

If $\log_2 x = -3$, what is the value of $x$?

Accepted Answer

$x = \frac{1}{8}$

Question 18

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

Accepted Answer

$\log_b a = \frac{\log_k a}{\log_k b}$

Question 19

If the demand function is $p = 4000 - 10x$ and $x = 40$, what is the total revenue $R(x)$?

Accepted Answer

$R(40) = (4000 - 10 	imes 40) 	imes 40 = 2400 	imes 40 = 96000$

QMS110 Lecture 9

Study Guide - Practice Questions

Study Guide - Flashcards