Download Properties of Logarithmic Functions Condensing Logarithmic

BGSU
3.3 Properties of Logarithms
Math 1300
Properties of Logarithmic Functions
Let b, M, N be positive real numbers with b 6= 1 and p and x are real numbers.
1. logb 1 = 0
2. logb b = 1
3. logb bx = x
4.blogb x = x, x > 0
5. logb M = logb N if and only if M = N
6. logb M N = logb M + logb N
7. logb M
N = logb M − logb N
8. logb M p = p logb M
Example 1 Use a calculator to evaluate each expression to three decimal places.
log 2
2
1. log
2. log 1.1
3. log 2 − log1.1
1.1
Example 2 Expand each logarithmic expression. 4
1. (#8) log9 x9
2. (#14) ln e8
4. (#32) log
q
5
x
y
5. (#34) logb
3. (#20) ln
√
3
xy 4
z5
√
7
x
6. (#38) ln
h
i
√
x4 x2 +3
5
(x+3)
Condensing Logarithmic Expression
Example 3 Write each expression in the following as a single logarithm whose coefficient is 1.
1. (#46) log3 405 − log3 5
2. (#54) 5 logb x + 6 logb y
3. (#62) 4 ln x + 7 ln y − 3 ln z
Ying-Ju Tessa Chen
Last modified: September 29, 2014
1
BGSU
3.3 Properties of Logarithms
4. (#66) 13 (log4 x − log4 y) + 2 log4 (x + 1)
Math 1300
5.(#68) 31 [5 ln(x + 6) − ln x − ln(x2 − 25)]
The Change-of-Base Expressions
For any logarithmic bases a and b, and any M > 0,
logb M =
loga M
.
loga b
Example 4 Use common logarithms or natural logarithms and a calculator to evaluate to four decimal
places.
2. (#76) log0.3 19
3. (#78) logπ 400
1. (#72) log6 17
Exercise 1 Expand each logarithm expression.
2. 99log99 3124750
1. log5 51024
4. logπ π 1770
5. 102 log10 4
3. log3 81
6. ln
e16
π 12
More exercises on the textbook: P. 449
Ying-Ju Tessa Chen
Last modified: September 29, 2014
2