BackStudy Notes: Logarithmic Functions (Precalculus, Section 3.2)
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Logarithmic Functions
Objectives
This section introduces the concept of logarithmic functions, their properties, and applications. Students will learn to convert between logarithmic and exponential forms, evaluate logarithms, use basic properties, graph logarithmic functions, determine domains, and work with common and natural logarithms.
Change from logarithmic to exponential form
Change from exponential to logarithmic form
Evaluate logarithms
Use basic logarithmic properties
Graph logarithmic functions
Find the domain of a logarithmic function
Use common logarithms
Use natural logarithms
Definition of the Logarithmic Function
Logarithmic Function with Base b
For x > 0 and b > 0, b ≠ 1:
Logarithmic form:
Exponential form:
The function is called the logarithmic function with base b.
Changing Between Logarithmic and Exponential Forms
Location of Base and Exponent
In , y is the exponent, b is the base.
In , b is the base, y is the exponent.
Examples
Logarithmic to Exponential:
means
means
means
Exponential to Logarithmic:
means
means
means
Evaluating Logarithms
Examples
because
because
because
because
Basic Logarithmic Properties
Properties Involving One
because
because
Examples
Inverse Properties of Logarithms
Examples
Graphs of Exponential and Logarithmic Functions
Table of Values
x | f(x) = 3^x |
|---|---|
-2 | |
-1 | |
0 | 1 |
1 | 3 |
x | g(x) = \log_3 x |
|---|---|
-2 | |
-1 | |
1 | 0 |
3 | 1 |
Graphical Features: is increasing and passes through (0,1). is increasing and passes through (1,0). The graphs are reflections of each other across the line .
Characteristics of Logarithmic Functions
Domain: (all positive real numbers)
Range: (all real numbers)
Intercept: Passes through (1, 0); no y-intercept
Increasing/Decreasing: If , is increasing; if , it is decreasing
Vertical Asymptote: (the y-axis)
Domain of a Logarithmic Function
The domain of consists of all for which .
Example
Find the domain of :
Domain:
Common Logarithms
The logarithmic function with base 10 is called the common logarithmic function. It is usually written as .
Properties of Common Logarithms
General Properties | Common Logarithms |
|---|---|
Application Example
The percentage of adult height attained by a boy who is years old can be modeled by , where is age (5 to 15) and is the percentage of adult height. For :
Natural Logarithms
The logarithmic function with base is called the natural logarithmic function. It is usually written as .
Properties of Natural Logarithms
General Properties | Natural Logarithms |
|---|---|
Application Example
The temperature increase in an enclosed vehicle after minutes is modeled by . For :
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