$$f(x) = a^x$$, where $$a \u003e 0$$ and $$a \neq 1$$

Question 1

Which of the following best describes the general form of an exponential function?

Accepted Answer

$$f(x) = a^x$$, where $$a \u003e 0$$ and $$a 
eq 1$$

Question 2

Suppose you invest $$1000 at an annual interest rate of 5\%$$ compounded yearly. What is the amount after 3 years?

Accepted Answer

$$1000(1.05)^3$$

Question 3

If a radioactive substance decays according to $$Q(t) = Q_0$ e$$^{-kt}$$, which of the following statements is true?

Accepted Answer

The quantity decreases exponentially over time.

Question 4

Which of the following is the correct formula for the half-life $$t_{1/2} of a $$substance decaying exponentially as $$Q(t) = Q_0$ e$$^{-kt}$$?

Accepted Answer

$$t_{1/2} = \frac{\ln 2}{k}$$

Question 5

Which of the following is the definition of the natural logarithm function?

Accepted Answer

$$\ln x = \log_e x$$

Question 6

Given $$f(x) = 2^x$$, which of the following is the inverse function $$f^{-1}(x$$)$$?

Accepted Answer

$$f^{-1}(x$$) = \$$log_2 x$

Question 7

Which property of logarithms allows you to write $$\log_a(xy) = \log_a x + \log_a y$$?

Accepted Answer

Product Rule

Question 8

If $$f(x) = e^x$$, which of the following statements is true about its graph?

Accepted Answer

It passes through the point $$(0,1)$$ and increases rapidly as $$x$$ increases.

Question 9

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

Accepted Answer

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

Question 10

A function $$f is $$one-to-one if and only if:

Accepted Answer

$$f(x_1$) = f(x_2$)$$ implies $$x_1$$ = $$x_2$$

Question 11

If $$f(x) = 3^x$$ and $$f^{-1}(x$$)$$ is its inverse, what is f$$^{-1}$$(27)$$?

Accepted Answer

$$3$$

Question 12

Which of the following best describes the domain of the logarithmic function $$\log_a x$$?

Accepted Answer

$$(0, \infty)$$

Question 13

Suppose $$P(t) = 500e$$^{0.04t}$$ models the population of a city in thousands, t$ years after 2020. What is the population in 2030?

Accepted Answer

$$500e^{0.4}$$

Question 14

Which of the following is NOT a property of logarithms?

Accepted Answer

$$\log_a(x+y) = \log_a x + \log_a y$$

Question 15

If $$f(x) = \ln x$$, what is the range of $$f$$?

Accepted Answer

$$(-\infty, \infty)$$

Question 16

If $$f(x) = 2^x$$ and $$g(x) = \log_2$ x$$, which statement is true?

Accepted Answer

$$f$$ and $$g$$ are inverse functions.

Question 17

A sample of a radioactive element decays to 25\% of its original amount in 10 years. What is the decay constant $$k in$$ $$Q(t) = Q_0$ e$$^{-kt}$$?

Accepted Answer

$$k = \frac{\ln 4}{10}$$

Question 18

Which of the following is the correct expression for the inverse of $$f(x) = \ln(x-2)$$?

Accepted Answer

$$f^{-1}(x) = e^x + 2$

Question 19

If $$\log_a b = c$$, which of the following is equivalent?

Accepted Answer

$$a^c = b$$

Exponential and Logarithmic Functions: Definitions, Properties, and Applications

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