Question 1

Which of the following is the correct definition of the natural logarithm $$\ln(x)$$ for $$x \u003e 0$$?

Accepted Answer

$$\ln(x) = \$$int_1$$^x \frac{1}{t} \, dt$$

Question 2

Given $$f(x) = \frac{1}{x}$$, what is its antiderivative on the domain $$x \u003e 0$$?

Accepted Answer

$$F(x) = \ln(x) + C$$

Question 3

Which of the following statements is TRUE about the relationship between the natural logarithm and the exponential function?

Accepted Answer

The natural logarithm is the inverse of the exponential function.

Question 4

Evaluate $$\$$int_1$$^e \frac{1}{x} \, dx.$$

Accepted Answer

$$1$$

Question 5

If $$f(x) = \ln(x)$$, what is $$f'(x)$$ for $$x \u003e 0$$?

Accepted Answer

$$f'(x) = \frac{1}{x}$$

Question 6

Which property of logarithms is illustrated by $$\ln(ab) = \ln(a) + \ln(b)$$?

Accepted Answer

Product property

Question 7

Suppose you want to find the derivative of $$y = \ln(x^2$ + 3x + 4). $$Which rule should you use?

Accepted Answer

Chain rule

Question 8

What is the value of $$e$$, the base of the natural logarithm, to four decimal places?

Accepted Answer

$$2.7183$$

Question 9

Which of the following integrals results in a logarithmic function?

Accepted Answer

$$\int \frac{1}{x} \, dx$$

Question 10

If $$g(x) = e^x$$, what is $$g^{-1}(x$$)$$?

Accepted Answer

$$\ln(x)$$

Question 11

Which of the following is the correct differentiation property for $$\ln(xy)$$?

Accepted Answer

$$\frac{d}{dx} \ln(xy) = \frac{1}{x} + \frac{1}{y} \frac{dy}{dx}$$

Question 12

A medical researcher models the concentration of a drug in the bloodstream as $$C(t) = C_0$ e$$^{-kt}$$. What is the time t$ when the concentration drops to half its initial value?

Accepted Answer

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

Question 13

Which of the following is the correct integral for $$\int 	an(x) \, dx$$?

Accepted Answer

$$-\ln|\cos(x)| + C$$

Question 14

If $$h(x) = \ln(x^2$)$$, which of the following is TRUE?

Accepted Answer

$$h(x) = 2\ln(x)$$

Question 15

A chemical reaction rate is modeled by $$R(t) = k \ln(t+1)$$ for $$t \u003e 0. $$What is $$\frac{dR}{dt}$$?

Accepted Answer

$$\frac{k}{t+1}$$

Question 16

Which of the following is the correct property for $$\ln\left(\frac{a}{b}\right)$$?

Accepted Answer

$$\ln(a) - \ln(b)$$

Question 17

Suppose $$y = \ln(x^2$ + 2x + 1). $$What is $$\frac{dy}{dx}$$?

Accepted Answer

$$\frac{2x + 2}{x^2$ + 2x + 1}$$

Question 18

Which of the following is the correct formula for logarithmic differentiation of $$y = (x^2$ + 3x + 4)^5$?

Accepted Answer

$$\frac{dy}{dx} = 5(x^2$ + 3x + 4)^4$ (2x + 3)$$

Question 19

If $$f(x) = \ln(x)$$ and $$g(x) = e^x$$, which of the following is TRUE?

Accepted Answer

$$f(g(x)) = x$$

Calculus Study Notes: The Natural Logarithm, Exponential Functions, and Logarithmic Differentiation

Study Guide - Practice Questions

Study Guide - Flashcards