# Lesson 9

What is a Logarithm?

### Problem 1

For each equation in the left column, find in the right column an exact or approximate value for the unknown exponent so that the equation is true.

### Problem 2

Here is a logarithmic expression: $$\log_{10}100$$.

1. How do we say the expression in words?
2. Explain in your own words what the expression means.
3. What is the value of this expression?

### Problem 3

The base 10 log table shows that the value of $$\log_{10} 50$$ is about 1.69897. Explain or show why it makes sense that the value is between 1 and 2.

### Problem 4

Here is a table of some logarithm values.

1. What is the approximate value of $$\log_{10}(400)$$?
2. What is the value of $$\log_{10}(1000)$$? Is this value approximate or exact? Explain how you know.
$$x$$ $$\log_{10} (x)$$
200 2.3010
300 2.4771
400 2.6021
500 2.6990
600 2.7782
700 2.8451
800 2.9031
900 2.9542
1,000 3

### Problem 5

What is the value of $$\log_{10}(1,\!000,\!000,\!000)$$? Explain how you know.

### Problem 6

A bank account balance, in dollars, is modeled by the equation $$f(t) = 1,\!000 \boldcdot (1.08)^t$$, where $$t$$ is time measured in years.

About how many years will it take for the account balance to double? Explain or show how you know.

### Solution

(From Unit 4, Lesson 8.)

### Problem 7

The graph shows the number of milligrams of a chemical in the body, $$d$$ days after it was first measured.

1. Explain what the point $$(1,2.5)$$ means in this situation.
2. Mark the point that represents the amount of medicine left in the body after 8 hours.

### Solution

(From Unit 4, Lesson 3.)

### Problem 8

The exponential function $$f$$ takes the value 10 when $$x = 1$$ and $$30$$ when $$x = 2$$

1. What is the $$y$$-intercept of $$f$$? Explain how you know.
2. What is an equation defining $$f$$?