# Lesson 11

Evaluating Logarithmic Expressions

### Problem 1

Select all expressions that are equal to $$\log_2 8$$.

A:

$$\log_5 20$$

B:

$$\log_5 125$$

C:

$$\log_{10} 100$$

D:

$$\log_{10} 1,\!000$$

E:

$$\log_3 27$$

F:

$$\log_{10} 0.001$$

### Problem 2

Which expression has a greater value: $$\log_{10} \frac {1}{100}$$ or $$\log_2 \frac {1}{8}$$? Explain how you know.

### Problem 3

Andre says that $$\log_{10}(55) = 1.5$$ because 55 is halfway between 10 and 100. Do you agree with Andre? Explain your reasoning.

### Problem 4

An exponential function is defined by $$k(x)= 15 \boldcdot 2^x$$.

1. Show that when $$x$$ increases from 1 to 1.25 and when it increases from 2.75 to 3, the value of $$k$$ grows by the same factor.
2. Show that when $$x$$ increases from $$t$$ to $$t+0.25$$, $$k(t)$$ also grows by this same factor.

### Solution

(From Unit 4, Lesson 5.)

### Problem 5

How many times does $1 need to double in value to become$1,000,000? Explain how you know.

### Solution

(From Unit 4, Lesson 8.)

### Problem 6

What values could replace the “?” in these equations to make them true?

1. $$\log_{10} 10,\!000 = {?}$$
2. $$\log_{10} 10,\!000,\!000 = {?}$$
3. $$\log_{10} {?} = 5$$
4. $$\log_{10} {?} = 1$$

### Solution

(From Unit 4, Lesson 9.)

### Problem 7

1. What value of $$t$$ would make the equation $$2^t = 6$$ true?
2. Between which two whole numbers is the value of $$\log_2 6$$? Explain how you know.

### Solution

(From Unit 4, Lesson 10.)

### Problem 8

For each exponential equation, write an equivalent equation in logarithmic form.

1. $$3^4 = 81$$
2. $$10^0 = 1$$
3. $$4^\frac12= 2$$
4. $$2^t = 5$$
5. $$m^n = C$$