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5.3 Logarithms
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Every exponential function has an inverse. The inverses of exponential functions are called
logarithmic functions (logarithms or logs for short).
Definition:
log b x = y means b y = x .
The functions f ( x ) = b and g ( x ) = log b x are inverses of each other.
x
b is called the base of the logarithm.
Example 1:
x
f
(
x
)
=
3
, then f (1) = 3 and f ( 2) = 9
Evaluate
We call this inverse function a logarithmic function and denote it f
So f
−1
−1
( x ) = log 3 x .
(3) = log 3 3 = 1 and f −1 (9) = log 3 9 = 2 .
y = log a x is called the logarithmic form. The form a y = x is the exponential form. You
should be able to go back and forth between the two forms.
Examples 2: Write each in the exponential form.
a. log4 x = 2
b. log 10,000 = 4
1
c. ln 4 = −4
e
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5.3 Logarithms
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Example 3: Write in the logarithmic form.
4
a. 3
b. 10
c. 2
d. e
= 81
3
−5
ln 3
= 1000
= . 03125
= 3
Remember we said e was a very important number? It is so important that the logarithmic function
of base e has its own special notation and its own button on your calculator.
The logarithm of base e is called the natural logarithm, which is abbreviated “ln”.
log e x = ln x
Evaluating logarithms:
Example 4: Evaluate log 2 8 .
In other words, log 2 8 is a number. What number is it?
This question is asking us to find a certain exponent. Specifically, “what exponent must I put on the
2 to give me 8?”
Said another way, “2 raised to what power is 8?”
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5.3 Logarithms
Examples 5: Simplify each of the following:
a. log 3 9
b. log 2
1
4
c. log 5
5
d. log 4 64
e. log 3 1
f. ln e
6
g. log 5 25
Example 6: Find log 2 ( −4) and log 5 0 .
IMPORTANT:
You cannot apply a logarithm to zero
or to a negative number!!!
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5.3 Logarithms
Facts about the graphs of y = b and y = log b x :
x
Actual graphs:
Example 7: Find the domain of f ( x ) = log 3 ( x − 4) .
Example 8: Find the domain of f ( x ) = log 5 ( x ) .
2
4
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5.3 Logarithms
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Example 9: Find the domain of g ( x ) = ln(3 − 2 x ) .
Example 10: Find the domain f(x) = log(4x + 3) + 5
Example 11: Graph the following natural log:
ln (x + 2 ) + 1
Example 12: Graph the following logarithms: f ( x ) = log 5 (x − 1) − 1
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5.3 Logarithms
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Example 13: Graph, state domain, range, asymptotes and the transformation of the key point
( 1, 0)
a. log 4 x
c. log(− x )
e. ln(x + 2 ) − 1
b. − log 2 x
d. log 3 (x − 1)