Q: Which of the following is the correct definition of an exponential function?

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

Q: Which of the following is the correct definition of a logarithmic function with base $$a$$?

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

Question 1

Which of the following is the correct definition of an exponential function?

Accepted Answer

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

Question 2

Which of the following statements about the domain and range of the exponential function $$f(x) = b^x ($$where $$b \u003e 0$$ and $$b 
eq 1) is $$correct?

Accepted Answer

Domain: $$x \in \mathbb{R}$$, Range: $$y \u003e 0$$

Question 3

Suppose a population of bacteria grows according to $$N = N_0$ e$$^{kt}$$, where N_0$ = 25$$ and $$k = 1.386. $$How many bacteria will there be after $$t = 0.6$$ hours? (Round to the nearest integer)

Accepted Answer

Approximately 57

Question 4

Which of the following is the correct formula for the amount $$A in an $$account after $$t$$ years if $$P is $$invested at an annual rate $$r$$ compounded $$m$$ times per year?

Accepted Answer

$$A = P(1 + \frac{r}{m}$$)^{mt}$$

Question 5

A principal of $$1,000 is $$invested at $$10\%$$ annual interest compounded monthly for $$10$$ years. What is the future value $$A$$? (Use $$A = P(1 + \frac{r}{m}$$)^{mt}$$)

Accepted Answer

$$A \approx 2,707.04$$

Question 6

Which of the following is the correct formula for continuous compound interest?

Accepted Answer

$$A = Pe$$^{rt}$$

Question 7

If $$f(x) = 3^x$$, what is $$f(\sqrt{2})$$?

Accepted Answer

$$3^{\sqrt{2}$$} \approx 4.73$$

Question 8

Which of the following statements about the graph of $$f(x) = b^x$$ for $$0 \u003c b \u003c 1 is $$true?

Accepted Answer

The function decreases rapidly as $$x$$ increases.

Question 9

Which of the following is the correct definition of a logarithmic function with base $$a$$?

Accepted Answer

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

Question 10

Convert the logarithmic equation $$\log_3$ x = 4 to $$exponential form and solve for $$x.$$

Accepted Answer

$$x = 81$$

Question 11

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

Accepted Answer

Product Rule

Question 12

Expand the expression $$\log_5$ \left(\frac{x^2$ y^3$}{z}\right)$$ using properties of logarithms.

Accepted Answer

$$2\log_5$ x + 3\log_5$ y - \log_5$ z$$

Question 13

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 14

If $$f(x) = \log_{10} x$$, what is the value of $$f(1000)$$?

Accepted Answer

$$f(1000) = 3$$

Question 15

Which of the following is the correct inverse function of $$f(x) = e^x$$?

Accepted Answer

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

Question 16

Given the law of forgetting $$\log P = \log P_0$ - c \log(t + 1)$$, if your initial score $$P_0 = 90$, c = 0.2$, and t = 2$ months, what is your expected score P$ after 2 months?

Accepted Answer

72

Question 17

Which of the following is the correct property of logarithms for $$\log_a(x^r)$$?

Accepted Answer

$$\log_a(x^r) = r \log_a x$$

Question 18

If $$f(x) = \log_2$ x$$, what is the domain and range of $$f(x)$$?

Accepted Answer

Domain: $$x \u003e 0$$, Range: $$y \in \mathbb{R}$$

Question 19

A demand function for a product is given by $$p = 3000 e$$^{-0.05q}$$, where q$ is the quantity demanded. What is the price per unit when q = 15$?

Accepted Answer

$$p \approx 406.01$$

QMS110 Lecture 8

Guida di studio - Domande di pratica

Study guide - Flashcard