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Exponential functions f (x) = ax are one-to-one functions. (from section 3.7) This means they each have an inverse
function. We denote the inverse function as loga, the logarithmic
function with base a.
5.3 Logarithmic Functions
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Definition
xayx ya log
Switch from logarithmic form to exponential form:
29log3
11.log10
3
12log8
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xayx ya log
Switch from exponential form to logarithmic form:
?49log7
2
116 4
1
?4log16
Evaluating logarithms
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x
y
xxf 2log)(
Create a table of points:
x
1/2
1
2
4
1
6
-1 1
-6
xy 2log
Graph
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x
yxxf alog)(
New domain restriction:1. No negative under an even root2. No division by zero3.
1
(1,0)
Domain:
Range:
Vertical Asymptote:
Graph
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1. loga1 = 0 (you must raise a to the power of 0 in order to get a 1)
2. logaa = 1 (you must raise a to the power of 1 to get an a)
3. logaax = x (you must raise a to the power of x to get ax)
4. alogax = x (logax is the power to which a must be raised to get x)
Rules 3 and 4 are results of the inverse property.#3 – putting the exponential function inside the logarithm function.#4 – putting the logarithm function inside the exponential function.
Properties
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xxxf loglog)( 10
With calculator:
Common Logarithm (Base 10)
Without calculator:
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xxxf e lnlog)(
With calculator:
Without calculator:
Natural Logarithm (Base e)