Solve Logarithmic Equation: Step-by-Step Guide
A logarithmic equation has the variable inside a log, like log_2(x) = 3. To solve it, use the log rules to combine everything into a single log, then switch to exponent form to get rid of the log. Always check that your answers keep everything inside every log positive, because you cannot take the log of zero or a negative number.
Math Racer skill guide · Algebra 2 · Grade 11
How to solve it, step by step
- Combine logs with the log rules. Use log(A) + log(B) = log(AB) and log(A) - log(B) = log(A/B) to write each side as one log.\[\log_b(A) + \log_b(B) = \log_b(AB)\]
- Get the log by itself. Move everything else away so the log is alone on one side.\[\log_b(f(x)) = c\]
- Switch to exponent form. Use the rule: log_b(y) = c means b^c = y. This removes the log.\[f(x) = b^{c}\]
- Solve for x. Solve the equation you get with the usual methods.\[f(x) - b^{c} = 0\]
- Check the domain. Put each answer into the original equation. Throw out any answer that makes something inside a log zero or negative.\[f(x_0) > 0 \;\text{required}\]
Worked example
Solve log_2(x) + log_2(x - 2) = 3.
Step 1: Combine the logs: log_2(x(x - 2)) = 3, which is log_2(x^2 - 2x) = 3.
Step 2: The log is already by itself.
Step 3: Switch to exponent form: x^2 - 2x = 2^3 = 8.
Step 4: Solve: x^2 - 2x - 8 = 0, which factors as (x - 4)(x + 2) = 0, so x = 4 or x = -2.
Step 5: Check the domain. The original equation needs x > 0 and x - 2 > 0, so x must be bigger than 2. Throw out x = -2 and keep x = 4.
Check: log_2(4) + log_2(2) = 2 + 1 = 3. Correct.
The solution is x = 4.
Tips and common mistakes
- Figure out what values are allowed first. Everything inside a log must be positive.
- Combine logs into one log BEFORE you switch to exponent form. You cannot do it one term at a time.
- Learn the three log rules: log(AB) = log A + log B, log(A/B) = log A - log B, and log(A^k) = k log A.
- log_b(y) = c is the same as b^c = y. This is the bridge between logs and exponents.
- If you get log_b(A) = log_b(B), then A = B (as long as both are positive).
- Fake answers are common. Always check in the original equation, not a later step.
- ln means log base e. All the same rules work.
- If the variable is in the base of the log, switch to exponent form or use change of base.