LOGARITHMIC FUNCTIONS

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SECTION 4.4

• LOGARITHMIC FUNCTIONS

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LOGARITHMIC FUNCTIONS
The logarithm (base b) of a number is the power
to which b must be raised to get that number.
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EXAMPLES
(a) log 4 16
2
(e) log 5 1
0
(b) log 3 27
3
(f) log 2 1/8
- 3
(g) log10 .1
- 1
(c) log e e 4
4
(d) log 2 32
5
(h) log 9 27
3/2
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LOGARITHMIC FUNCTIONS
Recall that only one-to-one functions have an
inverse. Exponential functions are
one-to-one. Their inverses are logarithmic
functions.
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LOGARITHMIC FUNCTIONS
Example Change the exponential expressions to
logarithmic expressions. 1.23 m eb 9 a4 24
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LOGARITHMIC FUNCTIONS
Example Change the logarithmic expressions to
exponential expressions. loga4 5 loge b -
3 log3 5 c
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DOMAIN OF A LOGARITHMIC FUNCTION
Since the logarithmic function is the inverse of
the exponential, the domain of a logarithmic is
the same as the range of the exponential.
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DOMAIN OF A LOGARITHMIC FUNCTION
Example Find the domain of the functions
below F(x) log2 (1 - x)
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SPECIAL LOGARITHMS
Logarithm to the base 10. Ex log 100 2
COMMON LOGARITHM
Logarithm to the base e. Ex ln e 2 2
NATURAL LOGARITHM
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THE NATURAL LOGARITHMIC FUNCTION

• Graph the function g(x) lnx in the same
  coordinate plane with f(x) ex
• Notice the symmetry with respect to the line y
  x.

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f(x) ex g(x) lnx
Compose the two functions g(f(x)) ln ex x
f(g(x)) eln x x We can see graphically as
well as algebraically that these two functions
are inverses of each other.
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Given f(x) bx Then f -1(x) log b x
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GRAPHS OF LOGARMITHMIC FUNCTIONS
1. The x-intercept is 1. 2. The y-axis is a
vertical asymptote of the graph. 3. A
logarithmic function is decreasing if 0 lt a lt
1 and increasing if a gt 1. 4. The graph is
continuous.
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GRAPHING LOGARITHMIC FUNCTIONS USING
TRANSFORMATIONS
Graph f(x) 3log(x 1). Determine the domain,
range, and vertical asymptote of f.
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EXAMPLE
Graph the function f(x) ln(1 - x). Determine
the domain, range, and vertical asymptote.
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SOLVING A LOGARITHMIC EQUATION
Solve log3(4x 7) 2 Solve logx64 2
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USING LOGARITHMS TO SOLVE EXPONENTIAL EQUATIONS
Solve e2x 5
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EXAMPLE
DO EXAMPLE 10 ON ALCOHOL AND DRIVING
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• CONCLUSION OF SECTION 4.4