Overview
This lecture covers how to solve logarithmic equations using key logarithm rules, including expanding, simplifying, and substitution, with a step-by-step worked example.
Logarithm Rules
- The logarithm of a fraction: log(a/b) = log(a) – log(b).
- The logarithm of a product: log(ab) = log(a) + log(b).
- The logarithm of a power: log(a^b) = b·log(a).
- The logarithm of a root: log(√x) = log(x^(1/2)) = (1/2)·log(x).
- All logarithms are base 10 in this context.
Worked Example (Solving the Equation)
- Expand log(100/x) as log(100) – log(x).
- Expand log(√x) as (1/2)·log(x).
- Expand log(10x^2) as log(10) + log(x^2) = log(10) + 2·log(x).
- After expanding, combine all log(x) terms.
- Use substitution: let u = log(x), rewrite all expressions in terms of u.
- Substitute known values: log(100) = 2, log(10) = 1.
- Combine like terms: 2 – u + 0.5u – 1 – 2u = 7/2 simplifies to –2.5u + 1 = 7/2.
- Isolate u: –2.5u = 5/2, so u = –1.
- Substitute back: log(x) = –1.
- Solve for x: x = 10^(–1) = 1/10.
Key Terms & Definitions
- Logarithm (log) — The exponent to which a base number must be raised to obtain a given number.
- Base 10 logarithm — A logarithm where the base is 10, denoted as log or lg.
Action Items / Next Steps
- Practice more logarithm equation problems using these rules.
- Review homework on logarithmic equations if assigned.