BackLogarithmic Functions: Definitions, Properties, and Applications
Study Guide - Smart Notes
Tailored notes based on your materials, expanded with key definitions, examples, and context.
Logarithmic Functions
Definition and Inverse Relationship
The logarithmic function is the inverse of the exponential function. It is commonly written as:
Definition: is the logarithm of with base .
This means: if and only if .
Exponential and Logarithmic Forms:
Exponential:
Logarithmic:
Example: because .
Converting Between Exponential and Logarithmic Forms
To convert to logarithmic form: .
To convert to exponential form: .
Example: can be written as .
Graphing Logarithmic Functions
Basic Graph and Properties
The graph of is the reflection of the graph of about the line .
The domain is .
The range is .
The function passes through the point because for any base .
There is a vertical asymptote at .
Example Table:
x | ||
|---|---|---|
-2 | 1/4 | (1/4, -2) |
-1 | 1/2 | (1/2, -1) |
0 | 1 | (1, 0) |
1 | 2 | (2, 1) |
2 | 4 | (4, 2) |
Graph: The graph of increases slowly for large and is undefined for .
Evaluating Logarithms
Solving Logarithmic Equations
To solve , rewrite as .
To solve , rewrite as .
Examples:
→ →
→ →
→ →
Try: →
Properties of Logarithmic Functions
Key Properties
(since )
(since )
(inverse property)
(inverse property)
(product property)
(quotient property)
(power property)
Natural logarithm: (where )
Examples:
(common logarithm, base 10)
Domain of Logarithmic Functions
Determining the Domain
The domain of a logarithmic function or is all such that .
Set the argument of the logarithm greater than zero and solve for .
Example: Find the domain of .
Set
Factor:
Solution: or
Domain:
Summary Table: Logarithmic Properties
Property | Equation | Description |
|---|---|---|
Product | Logarithm of a product is the sum of the logarithms. | |
Quotient | Logarithm of a quotient is the difference of the logarithms. | |
Power | Logarithm of a power is the exponent times the logarithm. | |
Inverse | Logarithm and exponential functions are inverses. | |
Change of Base | Allows computation of logarithms with any base. |
Additional info: The notes also reference the natural logarithm () and the common logarithm (), which are standard notations for logarithms with base and $10$, respectively.
