holt mcdougal algebra 2 properties of logarithms holt algebra 2holt mcdougal algebra 2 how do we use...
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Holt McDougal Algebra 2
Properties of LogarithmsProperties of Logarithms
Holt Algebra 2Holt McDougal Algebra 2
•How do we use properties to simplify logarithmic
expressions?
•How do we translate between logarithms in any base?
Holt McDougal Algebra 2
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.
Think: log j + log a + log m = log jam
Helpful Hint
Holt McDougal Algebra 2
Properties of LogarithmsAdding Logarithms
2
To add the logarithms, multiply the numbers.
1. log64 + log
69
log6 (4 9)
log6 36 Simplify.
Think: 6 x = 36.
Express as a single logarithm. Simplify, if possible.
= x
Holt McDougal Algebra 2
Properties of LogarithmsAdding Logarithms
6
To add the logarithms, multiply the numbers.
2. log5625 + log525
log5 (625 • 25)
log5 15,625 Simplify.
Think: 5 x = 15625.
Express as a single logarithm. Simplify, if possible.
= x
Holt McDougal Algebra 2
Properties of LogarithmsAdding Logarithms
–1
To add the logarithms, multiply the numbers.
Simplify.
Express as a single logarithm. Simplify, if possible.
= x
9
1log27log .3
3
1
3
1
9
127log
3
1
3log3
1
Think: x = 313
Holt McDougal Algebra 2
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.
Just as a5b3 cannot be simplified, logarithms must have the same base to be simplified.
Caution
Holt McDougal Algebra 2
Properties of LogarithmsSubtracting Logarithms
2
To subtract the logarithms, divide the numbers.
Simplify.
Express as a single logarithm. Simplify, if possible.
= x
4log100log .4 55
4
100log5
25log5
Think: 5 x = 25.
Holt McDougal Algebra 2
Properties of LogarithmsSubtracting Logarithms
1
To subtract the logarithms, divide the numbers.
Simplify.
Express as a single logarithm. Simplify, if possible.
= x
7log49log .5 77
7
49log7
7log7
Think: 7 x = 7.
Holt McDougal Algebra 2
Properties of Logarithms
Because you can multiply logarithms, you can also take powers of logarithms.
Holt McDougal Algebra 2
Properties of LogarithmsSimplifying Logarithms with Exponents
30
Powers expressed as multiplication.
Simplify.
Express as a product. Simplify, if possible.6
232log .6
32log6 2
56 Think: 2 x = 32.
Holt McDougal Algebra 2
Properties of LogarithmsSimplifying Logarithms with Exponents
10
Powers expressed as multiplication.
Simplify.
Express as a product. Simplify, if possible.20
16 4log .7
4log20 16
2
120 Think: 16
x = 4.
Holt McDougal Algebra 2
Properties of LogarithmsSimplifying Logarithms with Exponents
4
Powers expressed as multiplication.
Simplify.
Express as a product. Simplify, if possible.401log .8
10log4
14 Think: 10 x = 10.
Holt McDougal Algebra 2
Properties of LogarithmsSimplifying Logarithms with Exponents
4
Powers expressed as multiplication.
Simplify.
Express as a product. Simplify, if possible.2
5 25log .9
25log2 5
22 Think: 5 x = 25.
Holt McDougal Algebra 2
Properties of LogarithmsSimplifying Logarithms with Exponents
5
Powers expressed as multiplication.
Simplify.
Express as a product. Simplify, if possible.5
2 2
1log .10
2
1log5 2
15 Think: 2 x = ½ .
Holt McDougal Algebra 2
Properties of Logarithms
Exponential and logarithmic operations undo each other since they are inverse operations.
Holt McDougal Algebra 2
Properties of LogarithmsRecognizing Inverses
Simplify each expression.
12. log381 13. 5log510
11. log3311
log3311
11
log334
4
5log510
10
14. log100.9
Log10
100.9
0.9
15. 2log2(8x)
2log2(8x)
8x
Holt McDougal Algebra 2
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 LogarithmsChanging the Base of a Logarithm
16. Evaluate log32
8.
Change to base 10
log32
8 = log8log32 ≈ 0.6
17. Evaluate log927.
log927 = log27
log9 ≈ 1.5
Change to base 10
18. Evaluate log816.
log816 = log16
log8 ≈ 1.3
Change to base 10
Holt McDougal Algebra 2
Properties of Logarithms
Lesson 9.1 Practice A