# Lesson 11

Evaluating Logarithmic Expressions

- Let’s find some logs!

### 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 \(k(x)= 15 \boldcdot 2^x\).

- 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.
- Show that when \(x\) increases from \(t\) to \(t+0.25\), \(k(t)\) also grows by this same factor.

### Problem 5

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

### Problem 6

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

- \(\log_{10} 10,\!000 = {?}\)
- \(\log_{10} 10,\!000,\!000 = {?}\)
- \(\log_{10} {?} = 5\)
- \(\log_{10} {?} = 1\)

### Problem 7

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

### Problem 8

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

- \(3^4 = 81\)
- \(10^0 = 1\)
- \(4^\frac12= 2\)
- \(2^t = 5\)
- \(m^n = C\)