Properties
ofLogarithms
Logarithms
Properties of
•
•
How do we use properties to simplify logarithmic
expressions?
How do we translate between logarithms in any base?
HoltMcDougal
Algebra 2Algebra 2
Holt
Properties of Logarithms
The logarithmic function for pH that you saw in
the previous lessons, pH =–log[H+], can also be
expressed in exponential form, as 10–pH = [H+].
Because logarithms are exponents, you can derive
the properties of logarithms from the properties of
exponents
Holt McDougal Algebra 2
Properties of Logarithms
Remember that to multiply
powers with the same base,
you add exponents.
Holt McDougal Algebra 2
Properties of Logarithms
The property in the previous slide can be used in
reverse to write a sum of logarithms (exponents)
as a single logarithm, which can often be
simplified.
Helpful Hint
Think: log j + log a + log m = log jam
Holt McDougal Algebra 2
Properties of Logarithms
Adding Logarithms
Express as a single logarithm. Simplify, if possible.
1. log64 + log69
log6 (4  9)
To add the logarithms, multiply
the numbers.
log6 36 = x
Simplify.
2
Think: 6 x = 36.
Holt McDougal Algebra 2
Properties of Logarithms
Adding Logarithms
Express as a single logarithm. Simplify, if possible.
2. log5625 + log525
log5 (625 • 25)
To add the logarithms, multiply
the numbers.
log5 15,625 = x
Simplify.
6
Think: 5 x = 15625.
Holt McDougal Algebra 2
Properties of Logarithms
Adding Logarithms
Express as a single logarithm. Simplify, if possible.
1
3. log 1 27  log 1
3
3 9
 1
log 1 27  
9
3
log 1 3 = x
To add the logarithms, multiply
the numbers.
Simplify.
3
–1
Holt McDougal Algebra 2
Think:
1 x
3 =
3
Properties of Logarithms
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.
Holt McDougal Algebra 2
Properties of Logarithms
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.
Holt McDougal Algebra 2
Properties of Logarithms
Subtracting Logarithms
Express as a single logarithm. Simplify, if possible.
4. log 5100  log 5 4
100
log 5
4
To subtract the logarithms,
divide the numbers.
log 5 25 = x
Simplify.
2
Holt McDougal Algebra 2
Think: 5 x = 25.
Properties of Logarithms
Subtracting Logarithms
Express as a single logarithm. Simplify, if possible.
5. log 7 49  log 7 7
49
log 7
7
To subtract the logarithms,
divide the numbers.
log 7 7 = x
Simplify.
1
Holt McDougal Algebra 2
Think: 7 x = 7.
Properties of Logarithms
Because you can multiply logarithms, you can
also take powers of logarithms.
Holt McDougal Algebra 2
Properties of Logarithms
Simplifying Logarithms with Exponents
Express as a product. Simplify, if possible.
6. log 2 326
6 log 2 32
65
30
Holt McDougal Algebra 2
Powers expressed
as multiplication.
Think: 2 x = 32.
Simplify.
Properties of Logarithms
Simplifying Logarithms with Exponents
Express as a product. Simplify, if possible.
7. log 16 4
20
20 log 16 4
1
20 
2
10
Holt McDougal Algebra 2
Powers expressed
as multiplication.
Think: 16 x = 4.
Simplify.
Properties of Logarithms
Simplifying Logarithms with Exponents
Express as a product. Simplify, if possible.
8. log10 4
4 log 10
41
4
Holt McDougal Algebra 2
Powers expressed
as multiplication.
Think: 10 x = 10.
Simplify.
Properties of Logarithms
Simplifying Logarithms with Exponents
Express as a product. Simplify, if possible.
9. log 5 25
2
2 log 5 25
22
4
Holt McDougal Algebra 2
Powers expressed
as multiplication.
Think: 5 x = 25.
Simplify.
Properties of Logarithms
Simplifying Logarithms with Exponents
Express as a product. Simplify, if possible.
1
10. log 2  
2
5
1
5 log 2  
2
5 1
5
Holt McDougal Algebra 2
Powers expressed
as multiplication.
Think: 2 x = ½ .
Simplify.
Properties of Logarithms
Exponential and logarithmic operations undo each
other since they are inverse operations.
Holt McDougal Algebra 2
Properties of Logarithms
Recognizing Inverses
Simplify each expression.
12. log381
11
11. log33
13. 5log 10
5
log3311
log334
5log 10
11
4
10
14. log100.9
Log10100.9
0.9
Holt McDougal Algebra 2
5
15. 2log (8x)
2
2log (8x)
2
8x
Properties of Logarithms
Most calculators calculate logarithms only in base
10 or base e. You can change a logarithm in one
base to a logarithm in another base with the
following formula.
Holt McDougal Algebra 2
Properties of Logarithms
Changing the Base of a Logarithm
16. Evaluate log328.
17. Evaluate log927.
Change to base 10
Change to base 10
log328 =
log8
log32
≈ 0.6
18. Evaluate log816.
Change to base 10
log816 =
log16
log8
Holt McDougal Algebra 2
≈ 1.3
log927 =
log27
log9
≈ 1.5
Properties of Logarithms
Lesson 9.1 Practice A
Holt McDougal Algebra 2