Download 7-4 Properties of Logarithms

7-4 Properties of Logarithms
If two (or more) logs with the same base are added, you
can re-write them as a single log using multiplication.
Example)
log512 + log56 = log5(12•6) = log5(72)
log25x = log25 + log2x
Holt Algebra 2
7-4 Properties of Logarithms
Example 1: Adding Logarithms
Express log64 + log69 as a single logarithm.
Simplify.
log64 + log69
log6 (4  9)
To add the logarithms, multiply
the numbers.
log6 36
Simplify.
2
Think: 6? = 36.
Holt Algebra 2
7-4 Properties of Logarithms
Check It Out! Example 1b
Express as a single logarithm. Simplify, if possible.
log 1 27 + log 1
3
3
log 1 (27 •
3
log 1 3
1
9
)
1
9
To add the logarithms, multiply
the numbers.
Simplify.
3
–1
Holt Algebra 2
Think:
1 ?
3 =
3
7-4 Properties of Logarithms
If two (or more) logs with the same base are subtracted,
you can re-write them as a single log using division.
Example)
log512 – log56 =
13
log2( )
4
12
log5( )
6
= log213 – log24
Holt Algebra 2
= log52
7-4 Properties of Logarithms
Example 2: Subtracting Logarithms
Express log5100 – log54 as a single logarithm.
Simplify, if possible.
log5100 – log54
log5(100 ÷ 4)
To subtract the logarithms,
divide the numbers.
log525
Simplify.
2
Think: 5 = 25.
Holt Algebra 2
?
7-4 Properties of Logarithms
Check It Out! Example 2
Express log749 – log77 as a single logarithm.
Simplify, if possible.
log749 – log77
log7(49 ÷ 7)
To subtract the logarithms,
divide the numbers
log77
Simplify.
1
Holt Algebra 2
?
Think: 7 = 7.
7-4 Properties of Logarithms
If the argument of a log (what is inside the parenthesis)
is raised to a power, you can bring the power in front of
the log function.
Example)
log123
log(12•12•12)
log12 + log12 + log12
3log12
Note: log123 = log(123) = log(12)3 ≠ (log12)3
Holt Algebra 2
7-4 Properties of Logarithms
Example 3: Simplifying Logarithms with Exponents
Express as a product. Simplify, if possible.
a. log2326
b. log104
6log232
4log10
Because
6(5) = 30 25 = 32,
log232 = 5.
Holt Algebra 2
4(1) = 4
Because
101 = 10,
log 10 = 1.
7-4 Properties of Logarithms
Write as a single term.
a) log 3 − 2 log 5 + log 𝑥
3𝑥
log 2
5
b)
log2 3
log 8
+ 2
3
2
log 2 31
3
+ log 2 81
𝟑
3𝑥
log
25
Holt Algebra 2
log 𝟐
𝟑
𝟖
2
7-4 Properties of Logarithms
Write as multiple terms.
a)
3𝑥+𝑦
log 4 4
𝑧
b)
𝑦6
3log 5 2
𝑧
log 4 (3𝑥 + 𝑦) − log 4 𝑧 4
3 log 5 𝑦 6 − 3log 5 𝑧 2
log 4 (3𝑥 + 𝑦) − 4log 4 𝑧
18 log 5 𝑦 − 6log 5 𝑧
Holt Algebra 2
7-4 Properties of Logarithms
A logarithm with a base of “e” (the exponential 2.718...)
is called the natural log and is written as “ln.”
The equation ln(x) is read: “the natural log of x.”
The equation ln(2) is read: “the natural log of 2.”
Natural logs have all of the same properties as other
logs.
Holt Algebra 2
7-4 Properties of Logarithms
HW Logarithmic Worksheet on Homeroom
#’s 1 - 40
Holt Algebra 2