$$e = \lim_{n \to \infty} (1 + \frac{1}{n})^n$$

Question 1

Which of the following best describes the function $$f(x) = a^x$$ for $$a \u003e 1$$?

Accepted Answer

It is an increasing function for all $$x.$$

Question 2

What is the inverse function of $$f(x) = a^x$$?

Accepted Answer

$$f^{-1}(x$$) = \log_a(x)$$

Question 3

If $$2^x = 16$$, what is the value of $$x$$?

Accepted Answer

$$x = 4$$

Question 4

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

Accepted Answer

Product property

Question 5

Evaluate $$\log_3$(\sqrt{3})$$ exactly.

Accepted Answer

$$\frac{1}{2}$$

Question 6

Which of the following is true for the graph of $$y = a^x$$ when $$0 \u003c a \u003c 1$$?

Accepted Answer

The graph decreases as $$x$$ increases.

Question 7

Rewrite $$2^{-\frac{1}$${2}} = \frac{\sqrt{2}}{2}$$ as a logarithmic equation.

Accepted Answer

$$\log_2$(\frac{\sqrt{2}}{2}) = -\frac{1}{2}$$

Question 8

If $$\log_a(x) = y$$, which of the following is equivalent?

Accepted Answer

$$a^y = x$$

Question 9

Which property of logarithms is used to simplify $$\log_a\left(\frac{M}{N}\right)$$?

Accepted Answer

$$\log_a(M) - \log_a(N)$$

Question 10

Suppose $$f(x) = 3^x$$ and $$F(x) = \log_3$(x). $$What is $$F(f(x))$$?

Accepted Answer

$$x$$

Question 11

Which of the following is the correct definition of Euler’s number $$e$$?

Accepted Answer

$$e = \lim_{n 	o \infty} (1 + \frac{1}{n})^n$$

Question 12

If $$\log_5$(x) = 3$$, what is the value of $$x$$?

Accepted Answer

$$x = 125$$

Question 13

Which of the following is a property of logarithms?

Accepted Answer

$$\log_a(M^k) = k \log_a(M)$$

Question 14

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

Accepted Answer

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

Question 15

Which of the following statements is true about the function $$y = \log_a(x)$$ for $$a \u003e 1$$?

Accepted Answer

It is defined for $$x \u003e 0$$ only.

Question 16

If $$\log_2$(8) = x$$, what is $$x$$?

Accepted Answer

$$x = 3$$

Question 17

Which of the following is the correct relationship between exponentials and logarithms?

Accepted Answer

$$a^{\log_a(x)} = x$

Question 18

A medication decays in the bloodstream according to $$C(t) = C_0$ e$$^{-kt}$$. If the concentration halves every 3 hours, what is k$?

Accepted Answer

$$k = \frac{\ln 2}{-3}$$

Question 19

Which of the following is the correct simplification for $$\log_4$(16)$$?

Accepted Answer

$$2$$

Question 20

If $$\log_a(M) = p$$ and $$\log_a(N) = q$$, what is $$\log_a(MN)$$?

Accepted Answer

$$p + q$$

Exponentials and Logarithms: Definitions, Properties, and Applications

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