Solving Logarithmic Equations

Overview

The lecture covers step-by-step methods for solving simple logarithmic equations by converting them into exponential form and finding unknown values.

Converting Logarithmic to Exponential Form

• Logarithmic and exponential functions are interconnected and can be converted between forms.
• To solve a logarithmic equation, rewrite it in exponential form using the identity: log_b(a) = c ⇒ b^c = a.

Example Problems and Solutions

Problem 1: log₂(16) = x

• Convert to exponential: 2^x = 16.
• Express 16 as 2^4, so x = 4.

Problem 2: logₓ(27) = 3

• Convert to exponential: x^3 = 27.
• Express 27 as 3^3, so x = 3.

Problem 3: log₅(x) = 3

• Convert to exponential: 5^3 = x.
• Calculate: 5 × 5 × 5 = 125, so x = 125.

Problem 4: log₃₂(x) = 4/5

• Convert to exponential: 32^(4/5) = x.
• Express 32 as 2^5, so (2^5)^(4/5) = 2^4 = 16, so x = 16.

Problem 5: log₂(2x + 1) = 3

• Convert to exponential: 2^3 = 2x + 1.
• 2^3 = 8; rearrange: 8 - 1 = 2x ⇒ 7 = 2x ⇒ x = 7/2.

Checking Solutions

• Substitute the solution back into the original equation.
• Convert to exponential form and verify both sides are equal.

Key Terms & Definitions

• Logarithmic Equation — An equation that involves a logarithm with an unknown variable.
• Exponential Form — Rewriting a logarithmic equation as an exponential equation: log_b(a) = c ⇔ b^c = a.
• Exponent — Indicates how many times the base is multiplied by itself.

Action Items / Next Steps

• Practice converting logarithmic equations to exponential form and solving for unknowns.
• Watch the next video on applying laws of logarithms for more complex equations.