Logarithms Expansion Lecture Summary

Introduction

• Presenter: Paul
• Topic: Expanding logarithms using properties of logarithms
• Context: Lesson 10 in a tutorial series on logarithms

Key Concepts

Properties of Logarithms

1. Logarithm of a Fraction

   • Formula: ( \log_b \left( \frac{A}{B} \right) = \log_b(A) - \log_b(B) )
   • Application: Separate the logarithm of the numerator and the denominator.

2. Logarithm of a Product

   • Formula: ( \log_b(AB) = \log_b(A) + \log_b(B) )
   • Application: Split a product into the sum of logarithms.

3. Logarithm with an Exponent

   • Formula: ( \log_b(A^n) = n \cdot \log_b(A) )
   • Application: Bring the exponent out front as a coefficient.

Example Problem

Expression to Expand

• Given: ( \log_3 \left( \frac{x^2 y^3}{z} \right) )

Steps to Expand

1. Apply the Fraction Rule

   • ( \log_3(x^2 y^3) - \log_3(z) )

2. Apply the Product Rule to ( \log_3(x^2 y^3) )

   • ( \log_3(x^2) + \log_3(y^3) )

3. Apply the Exponent Rule

   • ( 2 \cdot \log_3(x) + 3 \cdot \log_3(y) )

4. Combining All Steps

   • Final Expansion: ( 2 \cdot \log_3(x) + 3 \cdot \log_3(y) - \log_3(z) )

Conclusion

• Effectively using properties of logarithms can simplify complex logarithmic expressions into a sum and difference form.
• Stay tuned for more examples and practice problems in upcoming lessons.
• Presenter encourages subscribing for more content.