Q: Given the exponential function $$f(x) = 42.2(1.56)^x$$, what is $$f(3)$$ rounded to the nearest cent?

$$f(3) = 42.2 \times (1.56)^3$ = 160.21$$

Question 1

Which of the following is the correct formula for the balance $$A in an $$account after $$t$$ years with principal $$P$$, annual interest rate $$r (as a $$decimal), and $$n$$ compounding periods per year?

Accepted Answer

$$A = P \left(1 + \frac{r}{n}\$$right)^{nt}$$

Question 2

Given the exponential function $$f(x) = 42.2(1.56)^x$$, what is $$f(3)$$ rounded to the nearest cent?

Accepted Answer

$$f(3) = 42.2 	imes (1.56)^3$ = 160.21$$

Question 3

Which transformation will shift the graph of $$f(x) = 3^x$$ left by 2 units?

Accepted Answer

$$g(x) = 3$$^{x+2}$$

Question 4

What is the domain of the logarithmic function $$f(x) = \log_{3}(x-5)$$?

Accepted Answer

$$x \u003e 5$$

Question 5

Rewrite the equation $$\log_{4} 26 = y in $$exponential form.

Accepted Answer

$$4^y = 26$$

Question 6

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

Accepted Answer

Product Rule

Question 7

Expand $$\log_2$(x^4$$ $$y^5$$)$ using logarithmic properties.

Accepted Answer

$$4 \log_2$ x + 5 \log_2$ y$$

Question 8

Condense $$2 \ln x + \frac{1}{3} \ln(x+5)$$ into a single logarithm.

Accepted Answer

$$\ln \left[x^2$ (x+5$$)^{1/3}$$\right]$$

Question 9

Solve for $$x$$: $$3^{2x-1} = 27$

Accepted Answer

$$x = 2$$

Question 10

If $$\log_{10} x = 2$$, what is the value of $$x$$?

Accepted Answer

$$x = 100$$

Question 11

Which of the following is the correct change-of-base formula for $$\log_5$ 20$$?

Accepted Answer

$$\log_5$ 20 = \frac{\log_{10} 20}{\log_{10} 5}$$

Question 12

A radioactive substance decays according to $$A = A_0$ e$$^{kt}$$, where k$$ \u003c 0$$. If the half-life is 21 years, what is the value of k$ (rounded to three decimal places)?

Accepted Answer

$$k = -0.033$$

Question 13

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

Accepted Answer

$$(0, \infty)$$

Question 14

The temperature $$T of a $$heated object at time $$t is $$given by Newton's Law of Cooling: $$T = C + (T_0$ - C)e$$^{kt}$$. What does C$ represent?

Accepted Answer

The temperature of the surrounding medium

Question 15

Which of the following is the correct equivalent of $$y = ab^x in $$base $$e$$?

Accepted Answer

$$y = a e$$^{x \ln b}$$

Question 16

A population grows according to $$P(t) = 800 e$$^{0.025t}$$. What is the population after 20 years (rounded to the nearest whole number)?

Accepted Answer

$$P(20) = 800 e$$^{0.5}$$ \approx 1314$$

Question 17

Which of the following is the correct logistic growth model formula?

Accepted Answer

$$f(t) = \frac{c}{1 + a e$$^{-bt}$$}$$

Question 18

If $$\ln x = 3$$, what is the value of $$x$$?

Accepted Answer

$$x = e^3$

Question 19

Which of the following statements about the graph of $$f(x) = a^x$$ for $$a \u003e 1 is $$true?

Accepted Answer

The graph passes through $$(0,1)$$ and increases as $$x$$ increases.

Chapter 3: Exponential and Logarithmic Functions – Precalculus Study Notes

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