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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: logb x = y means b
y
=x .
The functions f ( x ) = b x and g( x ) = log x are inverses of each other.
b
b is called the base of the logarithm.
Example 1:
Evaluate f ( x ) = 3 x , then f (1) = 3 and f (2) = 9
We call this inverse function a logarithmic function and
denote it f −1 ( x ) = log x .
b
So f −1 (3) = log3 3 = 1 and f −1 (9) = log 9 = 2 .
3
y = loga 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. log10 ,000 = 4
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c. ln
1
e
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5.3 Logarithms
2
= −4
Example 3: Write in the logarithmic form.
a. 34 = 81
b. 103 = 1000
c. 2 − 5 = . 03125
d. eln 3 = 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”.
loge = ln x
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5.3 Logarithms
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Evaluating logarithms:
Example 4:
Evaluate log2 8 .
In other words, log 8 is a number. What number is it?
2
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?”
Examples 5: Simplify each of the following:
a. log3 9
b. log2
1
4
c. log5 5
d. log4 64
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5.3 Logarithms
e. log3 1
f. ln e 6
g. log5 25
Example 6: Find log2 (−4 ) and log 0 .
5
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 x and y = logb x :
Actual graphs:
Example 7: Find the domain of f ( x ) = log3 ( x − 4) .
Example 8: Find the domain of f ( x ) = log5 ( x2 ) .
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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(4 x + 3) + 5
Example 11: Graph the following natural log:
ln(x + 2 ) + 1
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5.3 Logarithms
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Example 12: Graph the following logarithms: f ( x ) = log5 (x − 1) − 1
Example 13: Graph, state domain, range, asymptotes and the
transformation of the key point ( 1, 0)
a. log4 x
c. log(− x )
e. ln(x + 2 ) − 1
b. − log2 x
d. log3 (x − 1)