Warm Up

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Warm Up
Simplify.
1. (32)(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.
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Express as a single logarithm. Simplify, if
possible.
B. log5625 log525
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
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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 gt 0 and b?1 Example

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Example 4 Recognizing Inverses
Simplify each expression.
B. log381
A. log3311
C. 5log510
E. 2log2(8x)
D. log100.9
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
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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.