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Recipe

Solve a logarithmic equation

Use when: the unknown sits inside one or more logarithms

Goal: every exact value of the unknown, verified against the domain of the original equation

Taught in Solving exponential and logarithmic equations

Quick recipe

  1. Condense to a single log
  2. Rewrite in exponential form
  3. Solve the resulting equation
  4. Check every candidate against the ORIGINAL equation's domain
  1. 1

    If more than one logarithm appears on the same side, combine them into one using the product and quotient rules from earlier in this chapter.

    Example

    \log_3(x)+\log_3(x-6)=3 condenses to \log_3\big(x(x-6)\big)=3.

    Why: an equation still holding two or more separate logs can't be rewritten in exponential form in one move. Condensing first turns it into exactly the shape the next step needs.

  2. 2

    With a single \log_b(\text{expression})=c on one side, rewrite the whole equation using the definition of a logarithm: \text{expression}=b^c.

    Example

    \log_3\big(x(x-6)\big)=3 becomes x(x-6)=3^3=27.

    Why: this is exactly what \log_b(\text{expression})=c means, "b raised to c gives the expression." It's not a new move, just unwinding the definition from earlier in this chapter.

  3. 3

    What's left is an ordinary equation (often quadratic) with no logs remaining. Solve it with the usual tools.

    • it factors cleanly — factor and set each factor to zero.
    • it doesn't factor — use the quadratic formula.

    Why: condensing and rewriting removed every log, so nothing but ordinary algebra is left to finish the job.

  4. 4

    Substitute each candidate back into the equation as it was written before condensing, and discard any candidate that makes a logarithm's argument zero or negative.

    Why: condensing can enlarge the solution set; the domain of the condensed form isn't always the domain of the original. A candidate that solves the condensed equation can still be illegal in the equation actually being asked, so this check is not optional.

Common mistakes

  • Trying to rewrite in exponential form while two separate logs are still un-combined.
  • Checking a candidate against the condensed or exponential form instead of the original equation.
  • Throwing out a negative candidate on sight without checking whether it actually makes a log's argument \le0: the sign of x itself isn't the test, the sign of what's inside the log is.