Logarithms

1
Chapter 2 Exponents and Logarithms
2.5
Logarithms
2.5.1
MATHPOWERTM 12, WESTERN EDITION
2
Logarithmic Functions
A logarithmic function is the inverse of an
exponential function.
For the function y 2x, the inverse is x 2y.
In order to solve this inverse equation for y,
we write it in logarithmic form.
x 2y is written as y log2x
and is read as y the logarithm of x to base 2.
y 2x
2
4
8
1
16
2
4
8
1
16
y log2x
(x 2y)
2.5.2
3
Graphing the Logarithmic Function
y x
y 2x
y log2x
2.5.3
4
Comparing Exponential and Logarithmic Function
Graphs
y 2x
y log2x
The y-intercept is 1.
There is no y-intercept.
There is no x-intercept.
The x-intercept is 1.
The domain is x x Î R.
The domain is x x gt 0.
The range is y y Î R.
The range is y y gt 0.
There is a horizontal asymptote at y 0.
There is a vertical asymptote at x 0.
The graph of y 2x has been reflected in the
line of y x, to give the graph of y log2x.
2.5.4
5
Logarithms
Consider 72 49.
2 is the exponent of the power, to which 7 is
raised, to equal 49.
The logarithm of 49 to the base 7 is equal to 2
(log749 2).
Logarithmic form
Exponential notation
log749 2
72 49
In general If bx N,
then logbN x.
State in logarithmic form
State in exponential form
a) 63 216
log6216 3
a) log5125 3
53 125
b) log2128 7
27 128
b) 42 16
log416 2
2.5.5
6
Logarithms
State in logarithmic form
a)
b)
log2 32 3x 2
2.5.6
7
Evaluating Logarithms
1. log2128
2. log327
Note log2128 log227 7
log327 log333 3
log327 x 3x 27 3x 33
x 3
log2128 x 2x 128 2x 27
x 7
3. log556
logaam m
6
4. log816
5. log81
log816 x 8x 16 23x 24
3x 4
log81 x 8x 1 8x 80 x 0
loga1 0
2.5.7
8
Evaluating Logarithms
6.
7.
log4(log338)
x
log48 x 4x 8 22x 23 2x 3
2x 1
9. Given log165 x, and log84 y, express
log220 in terms of x and y.
8.
log165 x
log84 y
23 8
16x 5 24x 5
8y 4 23y 4
log220 log2(4 x 5) log2(23y x
24x) log2(23y 4x)
3y 4x
2.5.8
9
Evaluating Base 10 Logs
Base 10 logarithms are called common logs.
Using your calculator, evaluate to 3 decimal
places
a) log1025 b) log100.32
c) log102
1.398
-0.495
0.301
Evaluate log29
Change of base formula
log29 x 2x 9
log 2x log 9 xlog 2 log 9
x 3.170
2.5.9
10
Evaluating Logs
Given log3a 1.43 and log4b 1.86, determine
logba.
log3a 1.43 a 31.43
log a 1.43log 3
log4b 1.86 b 41.86
log b 1.86 log 4
logba 0.609
2.5.10
11
Assignment
Suggested Questions Pages 98-100 1-31
odd, 33-42, 47, 50 a, 52 a
2.5.11