IndietroExponential and Logarithmic Equations: Mini-Quiz Review
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Exponential Equations
Solving Exponential Equations with the Same Base
Exponential equations can often be solved by expressing both sides with the same base, then equating exponents.
Key Point: If , then (provided ).
Example: Solve .
Rewrite as :
Set exponents equal:
Solving Exponential Equations with Different Bases
Sometimes, bases must be rewritten to a common base before equating exponents.
Example:
Rewrite $27:
Set exponents equal:
Graphing Exponential Functions
Key Features of Exponential Graphs
Key Point: The general form is .
Key Features:
Key Point (KP): The y-intercept, found by setting .
Asymptote: The horizontal line that the graph approaches but never crosses.
Example:
KP:
Asymptote:
Logarithmic Equations
Solving Logarithmic Equations
Logarithmic equations can be solved by rewriting them in exponential form or by using properties of logarithms.
Key Point:
Example:
Rewrite in exponential form:
Example:
Solving Logarithmic Equations with Exponents
Example:
Rewrite $27:
Solving Equations with Rational Exponents
Solving for x with Rational Exponents
Equations involving rational exponents can be solved by raising both sides to the reciprocal power.
Key Point:
Example:
Raise both sides to the power:
Example:
Raise both sides to the power:
Summary Table: Key Properties of Exponential and Logarithmic Equations
Equation Type | Key Property | Solution Method |
|---|---|---|
Exponential (same base) | Rewrite bases, set exponents equal | |
Logarithmic | Rewrite in exponential form | |
Rational Exponents | Raise both sides to reciprocal power |
Additional info: The notes also include sketches of exponential graphs, highlighting key points and asymptotes, and emphasize the importance of rewriting expressions to common bases or forms for easier solving.
