Definition: An exponential equation has the unknown in an exponent, such as 4 to the x equals 50; it is solved by matching bases or by taking logarithms of both sides.
How do you recognize one?
The variable sits in the exponent: 23x – 1 = 32, or 6(1.05)t = 9. The ordinary tools of isolating a base-level variable do not work here.
What are the two strategies?
- Same base: rewrite both sides with one base, then set exponents equal.
- Logarithms: isolate the power, take a log of both sides, and bring down the exponent.
Worked example (same base): 23x – 1 = 32 = 25, so 3x – 1 = 5 and x = 2.
Worked example (logs): 6(1.05)t = 9. Divide by 6: (1.05)t = 1.5. Then t = ln 1.5 / ln 1.05 = 0.4055 / 0.04879, so t is about 8.31.
What should you watch for?
Do not take the log of each term separately before isolating the power. Mathbot also likes a quick substitution check at the end, since rounding can hide a slip in the algebra.
Example: (1.05)^t = 1.5 gives t = ln 1.5 / ln 1.05, about 8.31
Where to learn it
- Using Logs to Solve Exponent Equations – lesson
- Exponential Models: Growth, Decay, and Interest – lesson