$$\frac{1}{a} \tan$$^{-1}$$ \left(\frac{x}{a}\right) + C$$

Question 1

Which of the following statements correctly describes the relationship between logarithmic and exponential functions?

Accepted Answer

The logarithmic function is the inverse of the exponential function.

Question 2

Given $$y = \log_a x$$, which of the following is the equivalent exponential form?

Accepted Answer

$$x = a^y$$

Question 3

What is the domain and range of the function $$y = \log_a x$$ for $$a \u003e 1$$?

Accepted Answer

Domain: $$(0, \infty)$$, Range: $$(-\infty, \infty)$$

Question 4

Which property of logarithms allows you to write $$\log_a (AB) = \log_a A + \log_a B$$?

Accepted Answer

Product Property

Question 5

Expand the expression $$\log_5$ (x^3$ y^6$)$$ using the laws of logarithms.

Accepted Answer

$$3 \log_5$ x + 6 \log_5$ y$$

Question 6

Combine the expression $$3 \log x + 2 \log(x - 5)$$ into a single logarithm.

Accepted Answer

$$\log(x^3$ (x - 5)^2$)$$

Question 7

If $$\log_2$ 8 = 3$$, what is the equivalent exponential statement?

Accepted Answer

$$2^3 = 8$

Question 8

Which of the following is the correct derivative of $$y = e$$^{x^2}$$?

Accepted Answer

$$y' = 2x e$$^{x^2}$$

Question 9

What is the derivative of $$y = \ln(x^2$ + 1)$$?

Accepted Answer

$$y' = \frac{2x}{x^2$ + 1}$$

Question 10

Which of the following is the correct application of the chain rule for differentiating $$y = \ln(\sin x)$$?

Accepted Answer

$$y' = \frac{\cos x}{\sin x}$$

Question 11

Which of the following is the correct formula for logarithmic differentiation of $$y = x^x$$?

Accepted Answer

$$y' = x^x (1 + \ln x)$$

Question 12

What is the derivative of $$y = \sin$$^{-1}$$ x$$?

Accepted Answer

$$y' = \frac{1}{\sqrt{1 - x^2$}}$$

Question 13

Evaluate $$\sin$$^{-1}$$(\frac{\sqrt{3}}{2}).$$

Accepted Answer

$$\frac{\pi}{3}$$

Question 14

Which of the following integrals results in an inverse trigonometric function?

Accepted Answer

$$\int \frac{1}{\sqrt{1 - x^2$}} dx = \sin$$^{-1}$$ x + C$$

Question 15

Which property of logarithms allows you to write $$\log_a \left(\frac{A}{B}\right) = \log_a A - \log_a B$$?

Accepted Answer

Quotient Property

Question 16

If $$y = \cos$$^{-1}$$ x$$, what is the derivative $$\frac{dy}{dx}$$?

Accepted Answer

$$\frac{-1}{\sqrt{1 - x^2$}}$$

Question 17

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

Accepted Answer

$$\log_a x = \frac{\ln x}{\ln a}$$

Question 18

A biologist models population growth with $$P(t) = P_0$ e$$^{kt}$$. If the population doubles in 5 years, what is the value of k$?

Accepted Answer

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

Question 19

Which of the following is the correct integral for $$\int \frac{1}{a^2$ + x^2$} dx$$?

Accepted Answer

$$\frac{1}{a} 	an$$^{-1}$$ \left(\frac{x}{a}ight) + C$$

Logarithmic Functions, Logarithmic Differentiation, and Inverse Trigonometric Functions

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