Simplifying Logarithmic Expressions

Introduction

• Instructor: Joel
• Topic: How to simplify logarithmic expressions
• Purpose: Logarithms are used to undo variables in the exponent position, similar to how square roots undo squaring. They are especially useful for growth and decay problems.

Key Concepts

Properties of Logarithms

• Basic Property: Logarithm with the same base as the number cancels out the base.
  • Formula: ( \log_{b}(b^x) = x )
  • Example: ( \log_{3}(3^2) = 2 )

Identities

• Identity for One: ( \log_{b}(b) = 1 )
• Identity for Zero: ( \log_{b}(1) = 0 )

Simplification Process

• Approach: Rewrite the number above the base in terms of the base raised to an exponent.
• Cancel out: Simplify by canceling out the logarithm and base.

Examples

Easy Examples

1. Example 1: ( \log_{3}(9) )

   • Step: Rewrite 9 as (3^2)
   • Result: (2)

2. Example 2: ( \log_{4}(64) )

   • Step: Rewrite 64 as (4^3)
   • Result: (3)

More Involved Examples

1. Example 3: ( \log_{5}(\frac{1}{25}) )

   • Step: Rewrite (\frac{1}{25}) as (5^{-2})
   • Result: (-2)

2. Example 4: ( \log_{3}(9 \times \sqrt[4]{\frac{1}{27}}) )

   • Breakdown:
     • Rewrite 9 as (3^2)
     • Rewrite (\frac{1}{27}) as (3^{-3})
     • Fourth root: (\frac{\log_{3}((3^2) \times (3^{-3})^{1/4})}{\log_{3}(3^{5/4})})
   • Result: (5/4)

Conclusion

• Feedback: Questions or comments can be mailed to Joel at ytonline.com
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• Next Steps: Stay tuned for more lessons. Happy studying!