Logarithms: Step-by-Step Guide
A logarithm answers the question 'what power do I need?' log_b(x) = y means the same thing as b^y = x. For example, log_2(8) = 3 because 2^3 = 8. Logarithm rules turn multiplication into addition, division into subtraction, and exponents into multiplication. This makes them very useful for solving equations where the variable is in an exponent.
Math Racer skill guide · Algebra 2 · Grade 11
How to solve it, step by step
- Switch between log form and exponent form. log_b(x) = y is the same statement as b^y = x. Switching to exponent form is often the easiest way to find an answer.\[\log_{b}(x) = y \iff b^{y} = x\]
- Use the product, quotient, and power rules. Use these rules to combine or split logs that have the same base.\[\log_b(xy) = \log_b x + \log_b y,\;\; \log_b\!\left(\tfrac{x}{y}\right) = \log_b x - \log_b y,\;\; \log_b(x^{p}) = p\log_b x\]
- Use change of base if needed. If your calculator only has log base 10 or ln (base e), use the change-of-base formula to switch bases.\[\log_b x = \frac{\log_k x}{\log_k b}\]
- Solve or evaluate. After using the rules, either work out the number or get the log alone and switch to exponent form.\[\log_b 1 = 0,\quad \log_b b = 1\]
Worked example
Solve for x: log_2(x) + log_2(x - 2) = 3
Step 1: Combine the logs with the product rule: log_2(x(x - 2)) = 3.
Step 2: Switch to exponent form: x(x - 2) = 2^3 = 8.
Step 3: Multiply out: x^2 - 2x - 8 = 0.
Step 4: Factor: (x - 4)(x + 2) = 0, so x = 4 or x = -2.
Step 5: Throw out x = -2, because you cannot take the log of a negative number. The solution is x = 4.
Check: log_2(4) + log_2(2) = 2 + 1 = 3. Correct.
Tips and common mistakes
- You can only take the log of a positive number. The base must be positive and not equal to 1.
- Always check your answers in the original equation. The positive-number rule often rules out answers.
- Two useful facts: log_b(1) = 0, and log_b(b) = 1.
- 'log' with no base written usually means base 10. 'ln' means base e (the natural log).
- The product rule needs the SAME base on both logs. You cannot combine log_2 and log_3.
- The power rule moves exponents to the front: log(x^p) = p log(x). It is the key to solving exponent equations.
- Change of base lets you do any log on a basic calculator: log_b(x) = ln(x) / ln(b).