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Chapter 4 – Exponential and Logarithmic Functions. Section 4.3 Logarithmic Functions. Exponential Functions. Recall from last class that every exponential function f ( x ) = a x with a >0 and a 1 is a one-to-one function and therefore has an inverse function. - PowerPoint PPT Presentation
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4.3 - Logarithmic Functions
Section 4.3 Logarithmic Functions
Chapter 4 – Exponential and Logarithmic Functions
4.3 - Logarithmic Functions
Exponential Functions
Recall from last class that every exponential function f (x) = ax with a >0 and a 1 is a one-to-one function and therefore has an inverse function.
That inverse function is called the logarithmic function with base a and is denoted by loga.
4.3 - Logarithmic Functions
DefinitionLogarithmic Function
Let a be a positive number with a 1. The logarithmic function with base a, denoted by loga, is defined by
logax = y ay = x
So logax is the exponent to which the base a must be raised to give x.
4.3 - Logarithmic Functions
Switching Between Logs & Exp.
NOTE: logax is an exponent!
4.3 - Logarithmic Functions
Example – pg. 322 #5
Complete the table by expressing the logarithmic equation in exponential form or by expressing the exponential equation into logarithmic form.
4.3 - Logarithmic Functions
Properties of Logarithms
4.3 - Logarithmic Functions
Example – pg. 322Use the definition of the logarithmic function to find
x.
2 229. a) log 5 b) log 16
1 136. a) log 6 b) log 3
2 3x x
x x
4.3 - Logarithmic Functions
Graphs of Logarithmic Functions
Because the exponential and logarithmic functions are inverses with each other, we can learn about the logarithmic function from the exponential function. Remember,
Characteristic Exponential Logarithmic
Domain (-∞, ∞)
Range (0, ∞)
x-intercept None
y-intercept (0,1)
VA None
HA y = 0
4.3 - Logarithmic Functions
Graphs of Log Functions
4.3 - Logarithmic Functions
Example – pg. 323Graph the function, not by plotting points or using a
graphing calculator, but by starting from the graph of a logax function. State the domain, range, and asymptote.
53.
58.
2log 4f x x
3log 1 2y x
4.3 - Logarithmic Functions
DefinitionsCommon Logarithm
The logarithm with base 10 is called the common logarithm and is denoted by omitting the base:
Natural LogarithmThe logarithm with base e is called the natural logarithm and is denoted by:
10log logx x
ln logex x
4.3 - Logarithmic Functions
NoteBoth the common and natural logs can be evaluated
on your calculator.
4.3 - Logarithmic Functions
Properties of Natural Logs
4.3 - Logarithmic Functions
Example – pg. 322Find the domain of the function.
266. ( ) ln
67. = ln ln 2
g x x x
h x x x