AP Precalculus Section 2.13 Example: Rewriting a Literal Logarithmic Equation (Example 1)

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

Rewriting a literal logarithmic equation in exponential form involves applying the rules of logarithms, including the product, quotient, and power rules.

### Product Rule:
For two values \(a\) and \(b\) where \(a is greater than 0\), \(b is greater than 0\), and the same base \(b\):

- **Product Rule**: \( \log_b(ab) = \log_b(a) + \log_b(b) \)

### Quotient Rule:
For two values \(a\) and \(b\) where \(a is greater than 0\), \(b is greater than 0\), and the same base \(b\):

- **Quotient Rule**: \( \log_b\left(\frac{a}{b}\right) = \log_b(a) - \log_b(b) \)

### Power Rule:
For any value \(a\) where \(a is greater than 0\) and any real number \(n\) where \(n\) is not restricted, and the same base \(b\):

- **Power Rule**: \( \log_b(a^n) = n \cdot \log_b(a) \)

### Steps to Rewrite a Literal Logarithmic Equation in Exponential Form:

1. **Identify the Rules Used:** Determine which logarithmic rule (product, quotient, or power) is applied in the equation.

2. **Apply the Inverse Operation:** Use the inverse of the specific logarithmic rule to rewrite the equation in exponential form.

- If the equation involves a sum of logarithms, use the inverse of the product rule.
- If the equation involves the difference of logarithms, use the inverse of the quotient rule.
- If the equation involves an exponent on the argument of a logarithm, use the inverse of the power rule.

3. **Express in Exponential Form:** Convert the rewritten logarithmic equation into an exponential equation.

4. **Solve for the Variable:** Solve the exponential equation for the variable.

For instance, if you have the equation \( \log_b(x) + \log_b(y) = \log_b(xy) \), you're using the product rule. To rewrite it in exponential form:

1. Apply the inverse of the product rule: \( \log_b(x) + \log_b(y) = \log_b(xy) \) becomes \( xy = b^{\log_b(x) + \log_b(y)} \).
2. Utilize the properties of logarithms to simplify: \( xy = b^{\log_b(x) + \log_b(y)} = b^{\log_b(xy)} \).
3. Solve for \( xy = xy \).

By applying the appropriate inverse operation according to the logarithmic rule involved, you can convert the literal logarithmic equation into exponential form.

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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