Overview
This lecture covers the basics of exponential equations, methods to solve them, and provides worked examples including cases with different bases, fractions, roots, and equations resulting in 1.
Basic Form of Exponential Equations
- Exponential equations typically have the form ( a^{f(x)} = a^p ), where ( a ) is the base.
- If the bases are the same, set the exponents equal: ( f(x) = p ).
Solving When Bases Are the Same
- Example: ( 3^{3x-2} = 3^4 ) leads to ( 3x - 2 = 4 ); solve for ( x ).
- Substitute the found value of ( x ) back into the original equation to verify.
Making Bases the Same
- If bases differ, express both sides with the same base if possible (e.g., rewrite 8 as ( 2^3 )).
- Apply exponent rules when converting (e.g., ( (a^3)^{1/3} = a^{3 \times 1/3} = a^1 )).
Handling Fractions and Roots in Exponents
- Rewrite numbers like 1000 as ( 10^3 ), or ( 1/243 ) as ( 3^{-5} ).
- For fractional exponents/roots, use exponent rules: ( 1/a^n = a^{-n} ), and ( \sqrt[n]{a} = a^{1/n} ).
Solving Quadratic Equations from Exponents
- If an exponent equation leads to a quadratic (e.g., ( x^2 - 6x + 5 = 0 )), factor to find solutions for ( x ).
Dealing With Equations Equal to 1
- Recall: any base to the zero power is 1, ( a^0 = 1 ).
- Set the exponent equal to 0 and solve for ( x ).
Complex Forms: Division, Roots, and Multiple Steps
- When equations involve division or several exponent rules, simplify step by step to a basic exponential form.
- Multiply or divide to isolate the exponent expression.
Key Terms & Definitions
- Exponential Equation — An equation where variables appear as exponents.
- Base — The number being raised to a power in an exponent.
- Exponent Rules — Laws such as ( a^{m}/a^{n} = a^{m-n} ), ( (a^m)^n = a^{m \cdot n} ).
Action Items / Next Steps
- Practice solving exponential equations with both same and different bases.
- Review and apply exponent rules for simplifying complex equations.
- Factor quadratic equations as needed when solving for ( x ).