Lecture on Solving Logarithmic Equations

Introduction

• Goal: Learn to solve logarithmic equations efficiently
• Plan: Start with simple problems and progressively tackle harder ones
• Key focus: Handling natural logs and complex equations

Types of Logarithmic Equations

1. Equations with Logs on Both Sides

   • Example: log(4x - 5) = log(2x - 1)
     • Solve by setting arguments equal: 4x - 5 = 2x - 1
     • Steps:
       • Subtract 2x from both sides
       • Add 5 to both sides
       • Solution: x = 2

2. Equations with One Log Term

   • Example: -10 + log base 3 (x + 3) = -10
     • Isolate log: log base 3 (x + 3) = 0
     • Evaluate the log: 3^0 = 1
     • Solve: x = -2

Solving Techniques

1. Equations with All Log Terms

   • Example: log2(x + 2) + log2(x + 1) = log2(x) + log2(2x + 4)
     • Combine logs using addition/product property
     • Solve resulting quadratic: Check for extraneous solutions
     • Valid Solution: x = 1

2. Isolating Natural Logs

   • Example: 4ln(2x - 1) = 8
     • Isolate log: ln(2x - 1) = 2
     • Evaluate: e^2 = 2x - 1
     • Solve for x: x = (e^2 + 1) / 2

Complex Logarithmic Equations

1. Non-log Terms with Logs

   • Example: log2(x + 1) + log2(x - 1) = 3
     • Combine logs: log2((x + 1)(x - 1)) = 3
     • Evaluate: 2^3 = x^2 - 1
     • Solve: Check for extraneous solutions, valid x = 3

2. Handling Logs with Subtraction

   • Example: log(x + 10) - log(x - 1) = 2
     • Combine with division: log((x + 10)/(x - 1)) = 2
     • Evaluate with base 10: 10^2 = (x + 10)/(x - 1)
     • Solve: x = 10/9

Practice and Further Learning

• Try solving the sample problem: log(x) + log(x + 5) = log(6)
• Additional video available for more practice
• Access printable notes and additional resources via video description

Conclusion

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• Aim for continuous improvement and understanding of logarithmic equations.