Warm Up
2. (3–2)(35)214 33
38
1. (26)(28)
3. 4.
5. (73)5 715
44
Simplify.
Write in exponential form.
x0 = 16. logx x = 1 x1 = x 7. 0 = log
x1
Use properties to simplify logarithmic expressions.
Translate between logarithms in any base.
Objectives
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
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: logj + loga + logm = logjam
Helpful Hint
Express log64 + log
69 as a single logarithm.
Simplify.
Example 1: Adding Logarithms
2
To add the logarithms, multiply the numbers.
log64 + log
69
log6 (4 9)
log6 36 Simplify.
Think: 6? = 36.
Express as a single logarithm. Simplify, if possible.
6
To add the logarithms, multiply the numbers.
log5625 + log
525
log5 (625 • 25)
log5 15,625 Simplify.
Think: 5? = 15625
Check It Out! Example 1a
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.
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
Express log5100 – log
54 as a single logarithm.
Simplify, if possible.
Example 2: Subtracting Logarithms
To subtract the logarithms, divide the numbers.
log5100 – log
54
log5(100 ÷
4)
2
log525 Simplify.
Think: 5? = 25.
Express log749 – log
77 as a single logarithm.
Simplify, if possible.
To subtract the logarithms, divide the numbers
log749 – log
77
log7(49 ÷ 7)
1
log77 Simplify.
Think: 7? = 7.
Check It Out! Example 2
Express as a product. Simplify, if possible.
Example 3: Simplifying Logarithms with Exponents
A. log2326 B. log
8420
6log232
6(5) = 30
20log84
20( ) = 40 3
2 3
Because 8 = 4, log
84 = . 2
3
2 3
Because 25 = 32, log
232 = 5.
Express as a product. Simplify, if possibly.
a. log104 b. log5252
4log10
4(1) = 4
2log525
2(2) = 4Because 52 = 25, log
525
= 2.
Because 101 = 10, log
10 = 1.
Check It Out! Example 3
Example 4: Recognizing Inverses
Simplify each expression.
b. log381 c. 5log510a. log
3311
log3311
11
log33 3 3 3
log334
4
5log510
10
Most calculators calculate logarithms only in base 10 or base e (see Lesson 7-6). You can change a logarithm in one base to a logarithm in another base with the following formula.
Example 5: Changing the Base of a Logarithm
Evaluate log32
8.
Method 1 Change to base 10
log32
8 = log8log32
0.903 1.51
≈
≈ 0.6
Use a calculator.
Divide.
Example 5 Continued
Evaluate log32
8.
Method 2 Change to base 2, because both 32 and 8 are powers of 2.
= 0.6
log32
8 =
log28
log232
=3
5 Use a calculator.
Evaluate log927.
Method 1 Change to base 10.
log927 = log27
log9
1.4310.954
≈
≈ 1.5
Use a calculator.
Divide.
Check It Out! Example 5a
Evaluate log927.
Method 2 Change to base 3, because both 27 and 9 are powers of 3.
= 1.5
log927 =
log327
log39
=3
2 Use a calculator.
Check It Out! Example 5a Continued