1. 8-3 Logarithmic Functions as Inverses
Writing, graphing, and evaluating logarithmic expressions.

2. Objectives
Writing and Evaluating Logarithmic Expressions Graphing Logarithmic Functions

3. Vocabulary
The logarithm to the base b of a positive number y is defined as follows: If y = bx, then logb y = x

4. Example of Logarithm
Compare the amount of energy released in an earthquake that registers 6 on the Richter scale with one that registers 3.
Write a ratio.
E • 306
E • 303
= Simplify.
306
303
= 306–3 Division Property of Exponents
= 303 Simplify.
= 27,000 Use a calculator.
The first earthquake released about 27,000 times as much energy as the second.

5. Writing in Logarithmic Form
Write: 32 = 25 in logarithmic form.
If y = bx, then logb y = x. Write the definition.
If 32 = 25, then log2 32 = 5. Substitute.
The logarithmic form of 32 = 25 is log2 32 = 5.

6. Evaluating Logarithms
Evaluate log3 81.
Let log3 81 = x.
Log3 81 = x Write in logarithmic form.
81 = 3x Convert to exponential form.
34 = 3x Write each side using base 3.
4 = x Set the exponents equal to each other.
So log3 81 = 4.

7. Real-World Example
The pH of an apple is about 3.3 and that of a banana is about 5.2. Recall that the pH of a substance equals –log[H+], where [H+] is the concentration of hydrogen ions in each fruit. Which is more acidic?
Apple
Banana
pH = –log[H+]
pH = –log[H+]
3.3 = –log[H+]
5.2 = –log[H+]
log[H+] = –3.3
log[H+] = –5.2
[H+] = 10–3.3
[H+] = 10–5.2
5.0  10– 4
6.3  10– 6
The [H+] of the apple is about 5.0  10– 4.
The [H+] of the banana is about 6.3  10– 6.
The apple has a higher concentration of hydrogen ions, so it is more acidic.

8. Graphing a Logarithmic Function
Graph y = log4 x.
By definition of logarithm, y = log4 x is the inverse of y = 4x.
Step 1: Graph y = 4x.
Step 2: Draw y = x.
Step 3: Choose a few points on 4x. Reverse the coordinates and plot the points of y = log4 x.

9. Vocabulary Parent Function: Horizontal Reflection Stretch/Shrink
Vertical

10. Translating y = logbx Graph y = log5 (x – 1) + 2.
Step 1: Make a table of values for the parent function.
Step 2: Graph the function by shifting the points from the table to the right 1 unit and up 2 units. 1
125 25 5 –3 –2 –1 x
y = log5 x

11. Homework
8-3 Pg 449 & 450 # 6, 7, 14, 15, 38