Unit 3 · Test

Unit 3 test: Exponential and Logarithmic Functions

15 questions. Answer them all, then submit to see your score. Solutions unlock after you submit.

This test covers exponential functions and the number $e$ , logarithmic functions and log scales, properties of logarithms, exponential and logarithmic equations, and exponential and logistic models.

Question 1

Find the range of $h(x) = 4(0.5)^{x+2} - 1$ .

Enter an inequality, e.g. x >= 4, -2 < x <= 3, or x < 1 or x > 5

Question 2

Find the exponential function $y = ab^x$ whose graph passes through $(0, 6)$ and $(3, 48)$ .

Enter an expression, e.g. 3x^2 - 2x + 1

Question 3

You invest $3,000 at $5\%$ annual interest compounded continuously. What is the balance after $8$ years? Round to the nearest cent.

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Question 4

Evaluate $\log_2 \dfrac{1}{32}$ .

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Question 5

Find the domain of $f(x) = \ln(x + 2) - 3$ .

Enter an inequality, e.g. x >= 4, -2 < x <= 3, or x < 1 or x > 5

Question 6

A soft drink has a hydrogen-ion concentration of $\left[\text{H}^+\right] = 2.5 \times 10^{-4}$ moles per liter. Find its pH using $\text{pH} = -\log\left[\text{H}^+\right]$ . Round to the nearest hundredth.

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Question 7

Which is the expansion of $\log_3 \dfrac{9x^4}{y}$ ? Assume $x, y > 0$ .

Question 8

Evaluate $\log_4 2 + \log_4 32$ without a calculator.

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Question 9

Use the change-of-base formula to evaluate $\log_5 200$ . Round to three decimal places.

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Question 10

Solve $25^{x + 1} = 125^{x}$ .

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Question 11

Solve $6^{x - 2} = 40$ . Round to three decimal places.

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Question 12

Solve $\log_2(x + 6) + \log_2 x = 4$ .

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Question 13

A $120$ g sample of a radioactive substance decays to $45$ g in $10$ days. Find its half-life, rounded to the nearest hundredth of a day.

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Question 14

A population of rabbits on an island is modeled by $P(t) = \dfrac{900}{1 + 44e^{-0.5t}}$ , where $t$ is in years. When does the population reach half the carrying capacity? Round to the nearest hundredth of a year.

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Question 15

For the rabbit model $P(t) = \dfrac{900}{1 + 44e^{-0.5t}}$ , which statement is true?