Q: A bank pays $$7\%$$ interest annually. If you deposit $$\10$$,000$$, how much will you have after 10$ years? (Compound interest)

$$10,000 \times (1.07$$)^{10}$$

Q: At the start of a flu outbreak in a herd of cattle, there are $$5$$ infected individuals. The number doubles every $$3$$ days. How many days until there are $$80$$ infected cows?

$$n = 3 \times \frac{\log(80/5)}{\log(2)}$$

Question 1

Suppose you want to estimate $$\sqrt{70}$$ using logarithms. Which of the following steps correctly describes how to use logs to find $$\sqrt{70}$$?

Accepted Answer

Find $$\log(70)$$, divide by 2, then take the antilog.

Question 2

Computers use binary (base 2) rather than base 10. Given that $$\log(2) \approx 0.3$$, what is the approximate value of $$2^{10}$$?

Accepted Answer

$$1000$$

Question 3

If $$1 MB is $$really $$2^{20}$$ bytes, approximately how many bytes is $$1 MB $$using the log approximation $$\log(2) \approx 0.3$$?

Accepted Answer

$$1,000,000$$

Question 4

A population of rabbits doubles every generation. If there are $$20$$ generations, approximately how many times has the population multiplied?

Accepted Answer

$$2^{20}$$ \approx 1,000,000$$

Question 5

To solve the exponential equation $$10^x = 5$$, which of the following is the correct method?

Accepted Answer

Take $$\log of $$both sides: $$x = \log(5) / \log(10)$$

Question 6

Solve for $$x in $$the equation $$3^x = 7. $$What is the correct expression for $$x$$?

Accepted Answer

$$x = \log(7) / \log(3)$$

Question 7

Solve for $$x in $$the equation $$7^{x+2} = 30$. Which of the following is the correct solution?

Accepted Answer

$$x = \frac{\log(30)}{\log(7)} - 2$$

Question 8

Solve for $$y in $$the equation $$7 \cdot 3^y = 24$$^{y+3}$$. Which step is necessary to isolate y$?

Accepted Answer

Take $$\log of $$both sides and use the power rule.

Question 9

A bank pays $$7\%$$ interest annually. If you deposit $$\10$$,000$$, how much will you have after 10$ years? (Compound interest)

Accepted Answer

$$10,000 	imes (1.07$$)^{10}$$

Question 10

If you invest $$\1000$ at 14$$\%$$ annual interest, how much will you have after 5$ years?

Accepted Answer

$$1000 	imes (1.14)^5$

Question 11

If you invest $$\1000$ at 14$$\%$$ annual interest, how many years will it take to reach $$\$$7000$$?

Accepted Answer

$$n = \frac{\log(7)}{\log(1.14)}$$

Question 12

A population of bunnies triples every generation. If there are $$100$$ bunnies initially, how many bunnies are there after $$n$$ generations?

Accepted Answer

$$100 	imes 3^n$$

Question 13

How many generations does it take for the bunny population to reach $$107 if it $$triples each generation, starting from $$100$$?

Accepted Answer

$$n = \frac{\log(107/100)}{\log(3)}$$

Question 14

At the start of a flu outbreak in a herd of cattle, there are $$5$$ infected individuals. The number doubles every $$3$$ days. How many days until there are $$80$$ infected cows?

Accepted Answer

$$n = 3 	imes \frac{\log(80/5)}{\log(2)}$$

Question 15

Which of the following best describes the power rule for logarithms?

Accepted Answer

$$\log(a^p) = p \log(a)$$

Question 16

If you want to estimate $$2^{60}$$ using the log approximation $$\log(2) \approx 0.3$$, how many zeros will the answer have?

Accepted Answer

$$0.3 	imes 60 = 18$$ zeros

Question 17

Which of the following is a correct real-world application of exponential growth?

Accepted Answer

The spread of a virus in a population where each infected person infects two others every cycle.

Question 18

If you want to solve $$a^x = b$$ for $$x$$, which formula should you use?

Accepted Answer

$$x = \frac{\log(b)}{\log(a)}$$

Lecture 12: Exponents, Logarithms, and Their Applications in Calculus

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