Overview
This content focuses on solving the exponential equation (3(9^x) = 81), discussing solution strategies and key algebraic concepts commonly found in SAT, PSAT, and ACT math problems.
Solving the Exponential Equation (3(9^x) = 81)
- Divide both sides by 3 to isolate (9^x), resulting in (9^x = 27).
- Recognize that (9^x) can be rewritten as ((3^2)^x = 3^{2x}).
- Rewrite 27 as a power of 3: (27 = 3^3), so set (3^{2x} = 3^3).
- Set exponents equal to each other: (2x = 3).
- Solve for (x): (x = \frac{3}{2}).
Alternative Methods
- Some students suggest using logarithms to solve (9^x = 27), leading directly to (x = \frac{\log 27}{\log 9}).
- Different approaches can solve the equation, but rewriting both sides with the same base is often quickest.
Community Observations
- Multiple users agree the answer is (x = \frac{3}{2}) and emphasize the speed at which it can be solved.
- Comments note that the original solution might have used an unnecessarily complex approach despite simpler solutions being available.
- Some express confusion with the method, while others point out it's a standard problem taught at an early grade.
Key Terms & Definitions
- Exponential Equation — an equation where the variable is in the exponent, such as (a^{f(x)} = b).
- Base — the number that is raised to a power in an exponentiation (e.g., 3 in (3^x)).
- Logarithm — the inverse operation to exponentiation, used to solve for exponents.
Action Items / Next Steps
- Practice rewriting expressions with exponents to the same base for faster solutions.
- Try solving similar exponential equations using both exponent rules and logarithms.