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

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

Question 1

Which of the following best describes the inverse of the exponential function $$f(x) = a^x$$ for $$a \u003e 0$$ and $$a 
eq 1$$?

Accepted Answer

The inverse is $$f^{-1}(x$$) = \log_a(x)$$.

Question 2

If $$2^x = 16$$, what is the value of $$x$$ expressed as a logarithm?

Accepted Answer

$$x = \log_2$(16)$$

Question 3

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

Accepted Answer

Product property

Question 4

Given $$a \u003e 1$$, which graph best describes the function $$y = a^x$$?

Accepted Answer

An increasing curve passing through $$(0,1)$$

Question 5

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

Accepted Answer

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

Question 6

Which of the following is equivalent to $$\log_a\left(\frac{M}{N}\right)$$?

Accepted Answer

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

Question 7

If $$f(x) = 2^x$$, what is $$f^{-1}(8$$)$$?

Accepted Answer

$$3$$

Question 8

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

Accepted Answer

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

Question 9

Which property of logarithms allows you to write $$\log_a(M^k) = k \log_a(M)$$?

Accepted Answer

Power property

Question 10

If $$\log_2$(x) = -1$$, what is the value of $$x$$?

Accepted Answer

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

Question 11

A scientist models bacterial growth with $$N(t) = N_0$ e$$^{kt}$$, where N_0$ is the initial population and k$$ \u003e 0$$. If the population doubles in 3 hours, what is k$?

Accepted Answer

$$k = \frac{\ln(2)}{3}$$

Question 12

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

Accepted Answer

The function is decreasing for all $$x.$$

Question 13

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

Accepted Answer

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

Question 14

If $$\log_a(A) = x$$ and $$\log_a(B) = y$$, what is $$\log_a(AB)$$?

Accepted Answer

$$x + y$$

Question 15

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

Accepted Answer

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

Question 16

A medication decays in the bloodstream according to $$C(t) = C_0$ e$$^{-kt}$$. If the concentration halves every 5 hours, what is the value of k$?

Accepted Answer

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

Question 17

Which of the following is NOT a property of logarithms?

Accepted Answer

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

Question 18

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

Accepted Answer

$$125$$

Question 19

Which of the following is the correct expression for $$\log_a(1)$$ for any $$a \u003e 0$$, $$a 
eq 1$$?

Accepted Answer

$$0$$

Exponential and Logarithmic Functions: Definitions, Properties, and Applications

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