Solve exponential and logarithmic equations, checking for extraneous solutions, and apply these equations to real growth, decay, and half-life models.
When both sides of an exponential equation can be written with the SAME base, solving is simple: set the exponents equal and solve, since a one-to-one function like b^x hits each output only once. When the bases genuinely cannot be matched, take the logarithm of both sides and use the power property to pull the variable down out of the exponent.
A logarithmic equation solves by converting back to exponential form once the logarithm is isolated (or by condensing first, if a sum of logs appears). Because a logarithm is only defined for a positive input, every solution to a log equation must be checked against that requirement -- a candidate that would take the log of zero or a negative number is extraneous and must be rejected.
Mastery looks like: They solve exponential equations by matching bases when possible and by taking logarithms when not, solve logarithmic equations while rejecting extraneous solutions, and apply both to growth, decay, and half-life models.
Common stumbles: Trying to set exponents equal when the bases do not actually match; accepting a solution to a log equation without checking that every argument stays positive.
See it visually
Interactive, spoken-aloud tools that show this idea in motion.