1. LEARNING GOALS – LESSON 7.4 Example 1: Adding Logarithms
Warm Up
Simplify. 2.
1. (3–2)(35)
3. (73)5
Write in exponential form.
4. logx x = 1
5. 0 = logx1
LEARNING GOALS – LESSON 7.4
7.4.1: Use properties to simplify logarithmic expressions.
7.4.2: Translate between logarithms in any base.
Remember that to multiply powers with the same base, you add exponents.
Log Rule: Product Property
Example:
logbmn = logbm + logbn
log32x = log32 + log3x
Example 1: Adding Logarithms
A. Express log64 + log69 as a single logarithm. Simplify.

2. Example 2: Subtracting Logarithms
Express as a single logarithm. Simplify, if possible.
B. log log525
Just as a5b3 cannot be simplified, logarithms must have the same base to be simplified.
Caution
Remember that to divide powers with the same base, you subtract exponents
Log Rule: Quotient Property
Example:
Example 2: Subtracting Logarithms
Express as a single logarithm. Simplify, if possible.
A. log5100 – log54
B. log749 – log77

3. Example 3: Simplifying Logarithms with Exponents
Log Rule: Power Property
Example:
Example 3: Simplifying Logarithms with Exponents
Express as a product. Simplify, if possible.
A. log2326
B. log8420
C. log104
D. log5252
E. log2 ( ½ )5
Exponential and logarithmic operations undo each other
since they are ________________ operations.
INVERSE PROPERTIES
Log Rule: Inverse Properties
base b such that b > 0 and b≠1
Example:

4. Log Rule: Change of Base Example:
Example 4: Recognizing Inverses
Simplify each expression.
A. log3311
B. log381
C. 5log510
D. log100.9
E. 2log2(8x)
NOTE: YOUR CALCULATOR ONLY CAN COMPUTE LOGS IN BASE _____
and_____.
You can change a logarithm in one base to a logarithm in another base with the following formula.
Log Rule: Change of Base
Example:
*You may choose the a value
Example 5: Changing the Base of a Logarithm
Evaluate log328.
Method 2 Change to base 2, because both 32 and 8 are powers of 2.
Method 1 Change to base 10

5. Example 6: Geology Application
B. Evaluate log927.
Method 1 Change to base 10.
Method 2 Change to base 3, because both 27 and 9 are powers of 3.
C. Evaluate log816.
Method 2 Change to base _____, because both 27 and 9 are powers of _____.
Logarithmic scales are useful for measuring quantities that have a very wide range of values, such as the intensity (loudness) of a sound or the energy released by an earthquake. This is applicable because they go by powers of 10.
Example 6: Geology Application
The tsunami in December 2004 started by an earthquake with magnitude 9.3 How many times as much energy did this earthquake release compared to the 6.9-magnitude earthquake that struck San Francisco in1989?
Step 1: Plug in 9.3 and 6.9 for M.
Step 2: Solve for E.
Step 3: Compare your answers.