Question 1

Find the inverse of the function $$f(x) = 5 \cdot 10^x.$$

Accepted Answer

$$f^{-1}(x$$) = \log_{10}\left(\frac{x}{5}\right)$$

Question 2

Given $$f(x) = x + 1$$ and $$g(x) = \frac{1}{x-2}$$, find $$(f \circ g)(5).$$

Accepted Answer

$$\frac{4}{3}$$

Question 3

Solve for $$x$$: $$\log_{10}(x+3) - \log_{10}(x) = 1.$$

Accepted Answer

$$x = \frac{1}{3}$$

Question 4

Suppose $$y is $$increasing exponentially with $$y(0) = 5$$ and $$y(3) = 20. $$What is $$y(9)$$?

Accepted Answer

$$y(9) = 320$$

Question 5

Solve for $$x$$: $$\log_{10}(10$$^{2x}$$) = 12.$$

Accepted Answer

$$x = 6$$

Question 6

If $$250$$ dollars is deposited at $$2\%$$ interest compounded continuously, how long will it take to quadruple the money? (Round to two decimal places.)

Accepted Answer

$$t = 69.31$$ years

Question 7

Which equation correctly models the amount $$A(t) in an $$account with continuous compounding at rate $$k$$?

Accepted Answer

$$A(t) = A(0) \cdot e$$^{kt}$$

Question 8

A radioactive substance decays according to $$A(t) = A_0$ e$$^{-kt}$$. If the half-life is 20$ years, what is the value of k$? (Use $$\ln(2) \approx 0.693$$)

Accepted Answer

$$k = \frac{0.693}{20}$$

Question 9

If $$f(x) = 5 \cdot 10^x$$, what is the domain and range of $$f^{-1}(x$$)$$?

Accepted Answer

Domain: $$x \u003e 0$$, Range: $$(-\infty, \infty)$$

Question 10

A population grows according to $$P(t) = P_0$ e$$^{kt}$$. If the population doubles in 10$ years, how long will it take to triple?

Accepted Answer

$$t = 10 \cdot \frac{\ln(3)}{\ln(2)}$$

Question 11

Which of the following is equivalent to $$\log_{10}(a) + \log_{10}(b)$$?

Accepted Answer

$$\log_{10}(ab)$$

Question 12

If $$y = 5 \cdot 10^x$$, what is $$x in $$terms of $$y$$?

Accepted Answer

$$x = \log_{10}\left(\frac{y}{5}\right)$$

Question 13

A scientist measures a bacteria culture that quadruples every $$6$$ hours. If the initial population is $$1000$$, what is the population after $$18$$ hours?

Accepted Answer

$$64,000$$

Question 14

Solve for $$x$$: $$\log_{10}(x^2$) = 8.$$

Accepted Answer

$$x = 10^8$ or x$$ = $$-10^8$$

Question 15

If $$f(x) = x + 1$$ and $$g(x) = \frac{1}{x-2}$$, what is the domain of $$(f \circ g)(x)$$?

Accepted Answer

Question 16

A sum of money triples in $$t$$ years at a continuously compounded rate $$r. $$What is $$r in $$terms of $$t$$?

Accepted Answer

$$r = \frac{\ln(3)}{t}$$

Question 17

Which property of logarithms allows you to write $$\log_{10}\left(\frac{a}{b}\right) = \log_{10}(a) - \log_{10}(b)$$?

Accepted Answer

Quotient property

Question 18

If $$\log_{10}(x+3) - \log_{10}(x) = 1$$, what is the value of $$\frac{x+3}{x}$$?

Accepted Answer

$$10$$

Question 19

If $$y = 5 \cdot 10^x$$ models the intensity of a sound in decibels, and the intensity increases from $$y = 5 to$$ $$y = 50$$, by how much does $$x$$ increase?

Accepted Answer

$$x$$ increases by $$1$$

Precalculus Practice: Exponential, Logarithmic, and Function Operations

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