Section 4.2 ­ Logarithmic Functions ­ Inverse of Exponential Functions
The logarithmic function with base a (a≠1) is denoted by loga and is defined by logax = y if and only if ay = x
Examples:
log28 = 3
means 23 = 8
log525 = 2 means 52 = 25
log 100 = 2 means 102 = 100
You should be able to go back and forth
between logarithmic and exponential form.
convert to exponential form:
a. logab = c
b. log5125 = 3
convert to logarithmic form:
a. df = g
b. 24 = 16
Oct 2­7:06 AM
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Properties of Logarithms
1. loga1 =
2. logaa =
3. logaax =
logax
4. a =
Graphs of Logarithmic Functions ­ compare to exponential and notice change in domain in range
f(x) = 2x
f(x) = log2x
f­1(x) = logax, has a domain x > 0 and range R (all real numbers).
Oct 2­7:11 AM
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Graphs of Logarithmic Functions
The graph of the logarithmic function logax = y has:
1. Domain (0,∞) and range the set of all real numbers
2. a vertical asymptote at x = 0
3. an x­intercept at (1,0)
4. the following shape
y = logax
Family of logarithmic functions and shifts of logarithmic functions:
Compare the graphs of y = log2x, y = log3x , y = log4x
Compare the graph of y = log2x with g(x) = ­log2x and g(x) = log2(­x)
Shifts: Compare f(x) = log2x with
g(x) = 2 +log2x
and
h(x) = log2(x ­ 3)
Oct 2­7:16 AM
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Common Logarithm
The logarithm with base 10 is called the common logarithm
and is denoted by omitting the base: log x = log10x
Oct 2­7:24 AM
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Natural Logarithm
The logarithm with base e is called the natural logarithm and is denoted by ln: ln x = logex
ln x = y if and only if ey = x
You can shift, stretch, and reflect the natural logarithm just as you do regular logarithms. Domain is restricted by the vertical asymptote and the range is all Real Numbers.
Oct 2­10:31 AM
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