Logarithm Change of Base Formula and Solving Exponential Equations

Introduction

• Instructor: Jason (Math and Science Comm)
• Lesson Title: Logarithm Change of Base Formula and Solving Exponential and Logarithmic Equations

Importance of Logarithms

• Primarily used in solving equations, especially exponential equations.
• Logarithms can be complex with different bases (e.g., base 2, 3, 5, etc.).
• Difficulty arises when needing numerical answers and calculators lack specific base buttons.

Change of Base Formula

• A tool that allows computation of logarithms of any base using more common logarithms (base 10 or base e).
• Formula:
  [ \log_a(x) = \frac{\log_b(x)}{\log_b(a)} ]
  • Where:
    • ( a ) is the original base,
    • ( b ) is the new base,
    • ( x ) is the number being logged.

Understanding the Formula

• Base on the left can be changed to any base on the right.
• Example:
  • Change ( \log_3(9) ) to ( \frac{\log_2(9)}{\log_2(3)} ).
  • Can also change to base 10 or any other base.

Practical Examples

Example 1: Logarithm Calculation

• To calculate ( \log_4(7) ):
  • Use the change of base:
    [ \log_4(7) = \frac{\log_{10}(7)}{\log_{10}(4)} ]
  • Result: ( \approx 1.404 ) (rounded)

Example 2: Exponential Equation

• For ( 3^{2x} = 5 ):
  • Take logarithm of both sides:
    [ 2x = \log_3(5) ]
  • Use change of base:
    [ x = \frac{\log_{10}(5)}{2 \cdot \log_{10}(3)} ]
  • Calculate: ( x \approx 0.7326 ) (rounded)

Additional Examples

• Example: ( \log_{10}(x) = 0.8531853 )
  • Solve: ( x = 10^{0.8531853} \approx 7.13 ) (rounded)
• Example: ( 3^x = 30 )
  • Solve:
    [ x = \log_3(30) \approx \frac{\log_{10}(30)}{\log_{10}(3)} \approx 3.096 ]

Derivation of Change of Base Formula

1. Start with: ( \log_a(X) = Y ) implies ( a^Y = X ).
2. Take logarithm of both sides with base ( B ):
   [ \log_B(a^Y) = \log_B(X) ]
3. Rearranging gives:
   [ Y \cdot \log_B(a) = \log_B(X) ]
4. Solve for ( \log_a(X) ):
   [ \log_a(X) = \frac{\log_B(X)}{\log_B(a)} ]

Conclusion

• The change of base formula is crucial for solving logarithmic equations, especially with given bases unavailable on calculators.
• Understanding how to manipulate and derive the formula enhances problem-solving skills in algebra and calculus.