7.5: Apply Properties of Logarithms
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Objectives
Use properties to simplify logarithmic expressions.
Translate between logarithms in any base.
Common Core Standards:
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A-REI-2, F-BF-5
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Assessments:
Define all vocab for this section
Do worksheet 7-5
Remember that to multiply
powers with the same base,
you add exponents.
Expand the Logarithm
• Log7 6x
Express log64 + log69 as a single logarithm. Then Solve.
log64 + log69
log6 (4  9)
log6 36
2
To add the logarithms, multiply the numbers.
Simplify.
Think: 6? = 36.
Express as a single logarithm. Solve, if possible.
log5625 + log525
Given: Log 3= .477
What is log 21?
What is log 63?
Log 7= .845
The property above can also be used in reverse.
Caution
Just as a5b3 cannot be simplified, logarithms
must have the same base to be simplified.
Remember that to divide
powers with the same base,
you subtract exponents
Because logarithms are exponents, subtracting logarithms
with the same base is the same as finding the logarithms
of the quotient with that base.
Express log5100 – log54 as a single logarithm. Simplify, if
possible.
log5100 – log54
Express log749 – log77 as a single logarithm. Simplify, if
possible.
Given: Log 3= .477
What is log 7?
Log 21= 1.322
Because you can multiply logarithms, you can also
take powers of logarithms.
Power property of logarithms
Simplify
• Log3 1000
• Log2 x3
Express as a product. Simplify, if possible.
A. log2326
6log232
B. log8420
20log84
EXAMPLE 2
Expand a logarithmic expression
Expand log 6 5x3
y
SOLUTION
log 6 5x3 = log 6 5x3 – log 6 y
y
= log 6 5 + log 6 x3 – log 6 y
= log 6 5 + 3 log 6 x – log 6 y
Quotient property
Product property
Power property
Exponential and logarithmic operations undo each other since
they are inverse operations.
Simplify each expression.
a. log3311
b. log381
c. 5log510
a. Simplify log 100.9
b. Simplify 2log2(8x)
Most calculators calculate logarithms only in base 10 or base
e (see Lesson 7-4). You can change a logarithm in one base
to a logarithm in another base with the following formula.
• Change of Base formula
Loga b =
• Log3 8 =
log 8
log 3
log 𝑏
log 𝑎
=1.893
or
or
loga b =
ln 𝑎
ln 𝑏
log3 8 =
ln 8
ln 3
=1.893
Evaluate log328.
Evaluate log927.