Rewriting the Base
Part of the Exponentials, Logs & Sequences section of Coddy's Algebra 2 journey. Lesson 12 of 49.
In 27^x = 81, 81 is not a power of 27. But 27 and 81 are both powers of 3: 27 = 3^3 and 81 = 3^4. Rewrite both sides with base 3 and the equation becomes one you can match.
Writing 27 as 3^3 turns 27^x into (3^3)^x, and a power of a power multiplies the exponents: that is 3^(3x). So the equation reads 3^(3x) = 3^4. Equating the exponents gives 3x = 4, and x = 4/3. The answer does not have to be a whole number: a fraction is fine.
Challenge
Solve 4^x = 8. 4 and 8 are both powers of 2: write each side as a power of 2, then set the exponents equal.
Try it yourself
Solve for x
Test yourselfThis lesson includes a short quiz. Start the lesson to answer it and track your progress.
All lessons in Exponentials, Logs & Sequences
1Exponential Functions
Repeated MultiplyingGrowth or DecayDrawing GrowthDrawing DecayRecap: Exponential Functions3Same-Base Equations
Matching PowersRewriting the BaseAn Expression UpstairsFractions and NegativesRecap: Same Base