Solving Logarithm Equations

A logarithm equation is simply an equation where the unknown is inside a log, like $\log_2(x + 3) = 4$. Last lesson the unknown was stuck in an exponent and logs got it out; now it's the other way round, and exponentials get it out. The formula up top is just the definition of a log: $M$ (everything inside the log), $k$ (the value of the log) and $M = b^k$ (the same fact written as a power). That one switch does nearly all the work. Bear with the checking step at the end; it's the only real trap.

Two shapes. Most log equations come in one of these:

• One log equals a number: $\log_2(x + 3) = 4$ says "2 to the 4th is $x + 3$," so $x + 3 = 2^4$.
• Logs on both sides: $\ln A = \ln B$ means $A = B$, because a log never gives the same output for two different inputs.

If there are several logs on one side, use the log rules to combine them into one first.

Check for extraneous solutions. Here's why the checking matters. Combining $\ln x + \ln(x - 2)$ into $\ln(x(x - 2))$ quietly lets in negative $x$. At $x = -1$, the product $x(x - 2) = 3$ is positive, so the combined log is happy, but the original $\ln x$ and $\ln(x - 2)$ were never defined there. So reject any answer that makes an original argument zero or negative. These fake answers are called extraneous: the algebra produced them, but they don't solve the problem you started with.

Finding the original intensity behind a decibel or Richter reading, or the concentration behind a pH value, means solving a logarithm equation.