Overview
This lecture explains the importance of parentheses in logarithmic equations and demonstrates solving two similar-looking equations with different structures.
Understanding Parentheses in Logarithmic Expressions
- Parentheses determine whether the exponent applies to the argument of the logarithm or the entire logarithmic term.
- Without parentheses, the exponent applies only to the first term after "LG" (i.e., LG x² means log of x squared).
- With parentheses, the exponent applies to the entire logarithmic value (i.e., (LG x)² means log of x, then squared).
Solving Equation 1: LG x² – 3 LG x + 6 = 0
- LG x² = 2 LG x by logarithm rules.
- Combine terms: 2 LG x – 3 LG x + 6 = 0 becomes –LG x + 6 = 0.
- –LG x = –6; multiply both sides by –1 gives LG x = 6.
- Divide both sides by 3 (if needed), otherwise continue.
- Set both sides as exponents to base 10: x = 10² = 100.
Solving Equation 2: (LG x)² – 5 LG x + 6 = 0
- Use substitution: Let U = LG x, so equation becomes U² – 5U + 6 = 0.
- Solve quadratic: (U – 2)(U – 3) = 0, so U = 2 or U = 3.
- Convert back: LG x = 2 → x = 10² = 100; LG x = 3 → x = 10³ = 1000.
- Two solutions: x = 100 and x = 1000.
Key Terms & Definitions
- Logarithm (LG x) — The exponent to which the base 10 must be raised to get x.
- Quadratic Equation — An equation of the form ax² + bx + c = 0.
Action Items / Next Steps
- Practice distinguishing between LG x² and (LG x)² in problems.
- Review logarithm laws and their use in simplifying expressions.