Eleventh Grade · Math · week 29

Solving exponential and logarithmic equations

Solve exponential and logarithmic equations, checking for extraneous solutions from domain restrictions.

When two exponential expressions share the same base, solving is simple: set the exponents equal to each other and solve that ordinary equation, since a one-to-one function like b^x only hits each output once. But when the bases genuinely cannot be matched (like 5^x=20), you need a new move: take the logarithm of BOTH sides, then use the power property to pull the variable down out of the exponent.

Logarithmic equations solve by converting back to exponential form once the logarithm is isolated -- or, when a sum of logs shows up, by condensing first with last week's properties. Because a logarithm is only defined for a positive input, every solution to a log equation must be checked against that domain requirement; a candidate solution that would take the log of zero or a negative number must be rejected as extraneous.

📋 For the teacher
Teach it (10–15 min):
  • Ask "can these two sides share a base?" as the very first move on any exponential equation -- it decides whether logarithms are needed at all.
  • When logs are needed, narrate the power property step out loud every time: "the exponent walks down in front of the log" -- that single move is the whole new technique.
  • For log equations, insist on checking the domain (every argument of every original log must end up positive) before accepting any solution as final.
  • Show a same-base and a different-base example side by side so the decision point between the two methods is crystal clear.
Talk about it:
  • How do you decide whether an exponential equation needs logarithms to solve, or can be solved by matching bases?
  • Why does taking the log of both sides let you solve for a variable stuck in an exponent?
  • Why must a solution to a logarithmic equation be checked against the requirement that every log's argument be positive?

Mastery looks like: They solve exponential equations by matching bases when possible and by taking logarithms when not, and solve logarithmic equations while correctly rejecting any solution outside the domain.

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.

LogarithmsThe inverse of exponents — and what a log scale shows.try →ExponentsRepeated multiplication and how fast powers grow.try →
Homework for this lessonType it, write it with a stylus, or print it for pencil & paper.open →
Records attendance + 15 min of Math for this learner.
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