Overview
This lesson covers how to solve basic and slightly more advanced logarithmic equations by converting them to exponential form, applying logarithmic properties, and checking for extraneous solutions.
Converting Logarithmic to Exponential Form
- Logarithmic equations can be rewritten as exponential equations to solve for the unknown.
- Example: log₂16 = x becomes 2ˣ = 16; thus, x = 4.
Using Logarithmic Properties
- The change of base formula: logₐb = log b / log a.
- logₓ81 = 4 converts to x⁴ = 81, so x = 3 (fourth root of 81).
- log₅x = 3 converts to 5³ = x, so x = 125.
- For fractional exponents: log₃₂x = 4/5 leads to x = 32^(4/5) = 16.
Solving Equations with Variables Inside Logs
- log₃(5x+1) = 4 becomes 5x+1 = 81, x = 16.
- log x = 24 implies x = 10²⁴ (base 10 is assumed if not written).
- ln x = 7 leads to x = e⁷ ≈ 1096.65.
Quadratic Equations in Logarithms
- log₇(x²+3x+9) = 2 converts to x²+3x+9 = 49, solve as a quadratic.
- log₇x²+3x+9 = 2 → x = -8, 5 after factoring.
Combining Logarithms
- logₐ + log_b = log(ab); logₐ - log_b = log(a/b).
- log₂x + log₂(x+4) = 5 becomes log₂[x(x+4)] = 5, solve for x.
- Check for extraneous solutions; logs cannot have negative arguments.
Logarithmic Equations with Variables on Both Sides
- log₃(5x+2) = log₃(7x-8) leads to 5x+2=7x-8, x=5.
- log₂(x²+4x) = log₂5 gives x²+4x=5; solve for x.
Advanced Nested and Multi-step Logarithms
- log(log x) = 4 becomes log x = 10⁴ = 10,000, then x = 10^10,000.
- log₃(log₂x) = 2 means log₂x = 9, so x = 2⁹ = 512.
- log₁₀x² = log₁₀x² → x = 1 or x = 100 after factoring and solving.
Key Terms & Definitions
- Logarithm (logₐb) — The exponent x such that aˣ = b.
- Exponential form — Rewriting a log equation as aᵡ = b.
- Natural log (ln) — Logarithm with base e (~2.718).
- Change of base formula — logₐb = log b / log a.
- Extraneous solution — A solution that does not work in the original equation, such as negative logs.
Action Items / Next Steps
- Practice converting between logarithmic and exponential forms.
- Review properties of logarithms (product, quotient, and power rules).
- Complete any assigned homework problems on logarithmic equations.