AP Precalculus Section 2.12 Example: Combining Logarithms (Product and Power Properties)

Random AP Precalculus Problems (I found on the Internet). These are not official AP Collegeboard examples, but they will definitely get the job done!

The product and power properties are fundamental rules used when combining logarithms.

### Product Property:
The product property states that the logarithm of a product is equal to the sum of the logarithms of the individual numbers being multiplied.

For two positive numbers \( a \) and \( b \), with the same base \( b \) for the logarithms:

- \( \log_b(a \cdot b) = \log_b(a) + \log_b(b) \)

This property allows you to break down a logarithm of a product into the sum of logarithms of its factors.

### Power Property:
The power property deals with logarithms and exponents. It states that the logarithm of a number raised to an exponent is equal to the exponent multiplied by the logarithm of the base.

For a positive number \( a \), and any real number \( n \), with the same base \( b \) for the logarithms:

- \( \log_b(a^n) = n \cdot \log_b(a) \)

This property enables the simplification of logarithms involving exponents by bringing down the exponent as a coefficient.

### Using Both Properties Together:
When combining logarithms involving products and exponents, you can use these properties iteratively to simplify complex expressions. For example:

- \( \log_b(a \cdot b^n) \)
- Using the product property: \( \log_b(a) + \log_b(b^n) \)
- Then, applying the power property to \( \log_b(b^n) \): \( \log_b(a) + n \cdot \log_b(b) \)

These properties are incredibly useful in rearranging and simplifying logarithmic expressions, making calculations and problem-solving more manageable.

Remember to apply these properties carefully and ensure the bases of logarithms match when applying these rules.

The Topics covered in AP Precalculus are...

1.1 Change in Tandem
1.2 Rates of Change
1.3 Rates of Change in Linear and Quadratic Functions
1.4 Polynomial Functions and Rates of Change
1.5 Polynomial Functions and Complex Zeros
1.6 Polynomial Functions and End Behavior
1.7 Rational Functions and End Behavior
1.8 Rational Functions and Zeros
1.9 Rational Functions and Vertical Asymptotes
1.10 Rational Functions and Holes
1.11 Equivalent Representations of Polynomial and Rational Expressions
1.12 Transformations of Functions
1.13 Function Model Selection and Assumption Articulation
1.14 Function Model Construction and Application
2.1 Change in Arithmetic and Geometric Sequences
2.2 Change in Linear and Exponential Functions
2.3 Exponential Functions
2.4 Exponential Function Manipulation
2.5 Exponential Function Context and Data Modeling
2.6 Competing Function Model Validation
2.7 Composition of Functions
2.8 Inverse Functions
2.9 Logarithmic Expressions
2.10 Inverses of Exponential Functions
2.11 Logarithmic Functions
2.12 Logarithmic Function Manipulation
2.13 Exponential and Logarithmic Equations and Inequalities
2.14 Logarithmic Function Context and Data Modeling
2.15 Semi-log Plots
3.1 Periodic Phenomena
3.2 Sine, Cosine, and Tangent
3.3 Sine and Cosine Function Values
3.4 Sine and Cosine Function Graphs
3.5 Sinusoidal Functions
3.6 Sinusoidal Function Transformations
3.7 Sinusoidal Function Context and Data Modeling
3.8 The Tangent Function
3.9 Inverse Trigonometric Functions
3.10 Trigonometric Equations and Inequalities
3.11 The Secant, Cosecant, and Cotangent Functions
3.12 Equivalent Representations of Trigonometric Functions
3.13 Trigonometry and Polar Coordinates
3.14 Polar Function Graphs
3.15 Rates of Change in Polar Functions

I have many informative videos for Pre-Algebra, Algebra 1, Algebra 2, Geometry, Pre-Calculus, and Calculus. Please check it out:

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Nick Perich
Norristown Area High School
Norristown Area School District
Norristown, Pa

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