BackLecture 10: Logarithms: Properties, Evaluation, and Applications
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Logarithms
Definition and Basic Properties
Logarithms are fundamental mathematical tools used to solve equations involving exponentials. The logarithm logb(a) answers the question: "To what power must the base b be raised to produce a?"
Definition: If , then .
Common Logarithm: refers to logarithms with base 10, i.e., .
Notation: is shorthand for .
Evaluating Logarithms
To evaluate logarithms, use the definition and properties of exponents.
Example: Since , .
Example: Since , .
Example: Since , .
Estimating Logarithms
For numbers not easily expressed as powers of 10, logarithms can be estimated using tables or calculators.
Example: Estimate using a logarithm table or calculator.
Additional info: The table above shows logarithms to base 10 for numbers between 1.0 and 2.9, with increments of 0.01. To estimate , locate the row for 2.3 and column for 0.07 (since 2.37 = 2.3 + 0.07), then interpolate for 2.372.
Graphical Representation of Exponential Functions
The exponential function illustrates the relationship between exponents and their corresponding values. This graph helps visualize how logarithms invert the exponential process.
Additional info: The graph above shows the exponential growth of for between 0 and 1. Logarithms are the inverse of this function: for a given , .
Properties of Logarithms
Product Property
The product property allows the logarithm of a product to be expressed as the sum of logarithms:
Formula:
Example:
Applications Using Known Logarithms
Given a known logarithm, such as , other logarithms can be calculated using properties:
Example:
Example:
Example:
Example:
Using Logarithm Tables
Finding Logarithms and Powers of 10
Logarithm tables are useful for estimating logarithms and their inverses:
To find logarithms: Use the table forwards. For example, .
To find powers of 10: Use the table backwards. For example, .
Operation | Method | Example |
|---|---|---|
Find log of a number | Table forwards | |
Find power of 10 | Table backwards |
Additional info: Logarithm tables are especially useful before calculators, and for quick reference in exams or homework.
