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Table of Contents
• Recall an important property of inverse functions: the composite of the functions is x. If we assume that functions f and g are inverses of each other, then ...
x))x(g(f)x)(gf(
x))x(f(g)x)(fg(
• Since the exponential and logarithmic functions are inverses of each other, their composites result in x.
Logarithmic and Exponential Functions - Inverses
Table of Contents Slide 2
Logarithmic and Exponential Functions - Inverses
• Let exponential and logarithmic functions be given by: xb)x(f
xlog)x(g b
• Form the composite of f with g:xlogbb))x(g(f)x)(gf(
• Using the inverse property discussed earlier, this means ...
xb xlogb • Note that for the exponential and the logarithm, the bases are the same real number b.
Table of Contents Slide 3
Logarithmic and Exponential Functions - Inverses
• Example 1:
Simplify )2x(log33
Since this is the composite of an exponential function and a logarithmic function, each with a base of 3, the result is ...
2x3 )2x(log3
Table of Contents Slide 4
Logarithmic and Exponential Functions - Inverses
• Example 2:
Simplify )1xln(47e
)1xln(47e 4)1x(
1ln7
e
4)1x(
1ln
7ee 4
7
)1x(
1e
4
7
)1x(
e
4)1xln(7e
Table of Contents Slide 5
Logarithmic and Exponential Functions - Inverses
4e4 )1x(
1log
)1x(
1ln
ee
• Note the use of the inverse property in the next to last step where ...
4)1x(
1
Table of Contents Slide 6
Logarithmic and Exponential Functions - Inverses
• Let exponential and logarithmic functions be given by: xb)x(f
xlog)x(g b• Now form the composite of g with f:
xb blog))x(f(g)x)(fg(
• Using the inverse property g(f(x)) = x, this means ...
xblog xb
• Note that for the logarithm and the exponential, the bases are the same real number b.
Table of Contents Slide 7
Logarithmic and Exponential Functions - Inverses
• Example 3:
Simplify )3x2(4 4log
Since this is the composite of a logarithmic function and an exponential function, each with a base of 4, the result is ...
3x24log )3x2(4
Table of Contents Slide 8
Logarithmic and Exponential Functions - Inverses
• Example 4:
Simplify )e12ln( 7x
7xeln12ln
)7x(12ln
)e12ln( 7x
• Note the use of the inverse property in the last step where ...
7xe
7x elogeln 7x
Table of Contents
Logarithmic and Exponential Functions - Inverses
