Question 1

Which of the following correctly expresses the logarithm of a product using the properties of logarithms?

Accepted Answer

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

Question 2

Expand the expression $$\log_5$\left(\dfrac{\sqrt{x}}{25y^3$}\right)$$ using properties of logarithms.

Accepted Answer

$$\dfrac{1}{2}\log_5$(x) - 2 - 3\log_5$(y)$$

Question 3

Which of the following is NOT a correct application of the properties of logarithms?

Accepted Answer

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

Question 4

Condense the expression $$2\ln(x) + \dfrac{1}{3}\ln(x+5)$$ into a single logarithm.

Accepted Answer

$$\ln\$$left(x^2$$\sqrt[3]{x+5}\right)$$

Question 5

Expand $$\log_b\$$left(x^4$$\sqrt[3]{y}\right)$$ using properties of logarithms.

Accepted Answer

$$4\log_b(x) + \dfrac{1}{3}\log_b(y)$$

Question 6

Which of the following is the correct change-of-base formula for $$\log_7$(50)$$?

Accepted Answer

$$\dfrac{\ln(50)}{\ln(7)}$$

Question 7

A scientist is analyzing the pH of a solution, which is defined as $$\mathrm{pH} = -\log_{10}[H^+]. If $$the hydrogen ion concentration is multiplied by 100, how does the pH change?

Accepted Answer

It decreases by 2

Question 8

Condense the expression $$\dfrac{1}{4}\log_b(x) - 2\log_b(5) - 10\log_b(y)$$ into a single logarithm.

Accepted Answer

$$\log_b\left(\dfrac{\sqrt[4]{x}}{25y$$^{10}$$}\right)$$

Question 9

Which of the following is a common mistake when working with logarithms?

Accepted Answer

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

Question 10

Expand $$\ln\left(\dfrac{e^5$}{11}\right)$$ using properties of logarithms.

Accepted Answer

$$5 - \ln(11)$$

Question 11

Condense $$2\log(x-3) - \log(x)$$ into a single logarithm.

Accepted Answer

$$\log\left(\dfrac{(x-3)^2$}{x}\right)$$

Question 12

Expand $$\log_6$(3^9$$)$ using the power rule.

Accepted Answer

$$9\log_6$(3)$$

Question 13

Which of the following is the correct way to write $$\log_4$(7 	imes 5) as a $$sum?

Accepted Answer

$$\log_4$(7) + \log_4$(5)$$

Question 14

A financial analyst uses the formula $$A = P e$$^{rt}$$ to model compound interest. If A = 2P$, solve for t$ in terms of r$ using logarithms.

Accepted Answer

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

Question 15

Which of the following is the correct expansion of $$\log(x^2$(x+1)^3$$)$?

Accepted Answer

$$2\log(x) + 3\log(x+1)$$

Question 16

Which of the following is the correct way to write $$\log_b\left(\dfrac{M}{N}\right)$$ using properties of logarithms?

Accepted Answer

$$\log_b(M) - \log_b(N)$$

Question 17

Expand $$\log(x^2$\sqrt{y})$$ using properties of logarithms.

Accepted Answer

$$2\log(x) + \dfrac{1}{2}\log(y)$$

Question 18

Condense $$\log(25) + \log(4)$$ into a single logarithm and evaluate.

Accepted Answer

$$2$$

Question 19

Which of the following is the correct change-of-base formula for $$\log_3$(20)$$ using common logarithms?

Accepted Answer

$$\dfrac{\log(20)}{\log(3)}$$

Properties of Logarithms: Rules, Expansions, and Change of Base

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