# Lesson 8

Unknown Exponents

• Let’s find unknown exponents.

### Problem 1

A pattern of dots grows exponentially. The table shows the number of dots at each step of the pattern.

 step number number of dots 0 1 2 3 1 5 25 125
1. Write an equation to represent the relationship between the step number, $$n$$, and the number of dots, $$y$$.
2. At one step, there are 9,765,625 dots in the pattern. At what step number will that happen? Explain how you know.

### Problem 2

A bacteria population is modeled by the equation $$p(h) = 10,\!000 \boldcdot 2^h$$, where $$h$$ is the number of hours since the population was measured.

About how long will it take for the population to reach 100,000? Explain your reasoning.

### Problem 3

Complete the table.

 $$x$$ $$10^x$$ -2 0 $$\frac{1}{3}$$ 1 $$\frac{1}{10,000}$$ $$\frac{1}{1,000}$$ $$\frac{1}{100}$$ $$\hspace{.6cm}$$ $$\hspace{.6cm}$$ $$\hspace{.6cm}$$ 1,000 1,000,000,000

### Problem 4

Here is a graph of $$y = 3^x$$.

What is the approximate value of $$x$$ satisfying $$3^x = 10,\!000$$? Explain how you know.

### Problem 5

One account doubles every 2 years. A second account triples every 3 years. Assuming the accounts start with the same amount of money, which account is growing more rapidly?

### Problem 6

How would you describe the output of this graph for:

1. inputs from 0 to 1
2. inputs from 3 to 4
(From Unit 4, Lesson 1.)

### Problem 7

The half-life of carbon-14 is about 5730 years.

1. Complete the table, which shows the amount of carbon-14 remaining in a plant fossil at the different times since the plant died.
2. About how many years will it be until there is 0.1 picogram of carbon-14 remaining in the fossil? Explain how you know.
years picograms
0 3
5730
$$2 \boldcdot 5730$$
$$3 \boldcdot 5730$$
$$4 \boldcdot 5730$$
(From Unit 4, Lesson 7.)