Lecture on Logarithms

Introduction to Logarithms

• Definition: A logarithm is an exponent.
• Important to understand the relationship between logarithmic and exponential forms.
  • Logarithmic Form: ( \log_b{x} = y )
  • Exponential Form: ( b^y = x )
  • Example: ( \log_3{9} = 2 ) translates to ( 3^2 = 9 ).

Converting Between Forms

• Logarithmic to Exponential:
  • Example: ( \log_{10}{\frac{1}{100}} = -2 ) translates to ( 10^{-2} = \frac{1}{100} ).
• Exponential to Logarithmic:
  • Example: ( \log_5{125} = 3 ) translates back to ( 5^3 = 125 )._

Evaluating Logarithmic Expressions

• Example 1:
  • ( \log_2{64} = ? )
  • Express 64 as powers of 2: ( 2^6 = 64 )
  • Conclusion: ( \log_2{64} = 6 ).
• Example 2:
  • ( \log_3{243} = ? )
  • Express 243 as powers of 3: ( 3^5 = 243 )
  • Conclusion: ( \log_3{243} = 5 ).

Inverse Property of Exponents and Logarithms

• Logarithmic and exponential functions are inverses.
• Properties:
  • ( b^{\log_b{x}} = x )
  • ( \log_b{b^x} = x )
• Example:
  • ( \log_9{9^2} ) simplifies to 2.
  • ( \log_7{7^x} ) simplifies to ( x^2 - 1 ).

Properties of Logarithms

Addition (Product Rule)

• If multiplying inside the log, expand by addition.
  • Example: ( \log_2{(5 \times 4)} = \log_2{5} + \log_2{4} ).
• Expansion:
  • ( \log_{10}{(3xy)} = \log_{10}{3} + \log_{10}{x} + \log_{10}{y} ).
• Condensing:
  • Combine logs by multiplying: ( \log_3{8xyz} ).

Subtraction (Quotient Rule)

• If dividing inside the log, expand by subtraction.
  • Example: ( \log_7{\frac{1}{4}} = \log_7{1} - \log_7{4} ).
• Expansion:
  • ( \log_4{\frac{x}{8}} = \log_4{x} - \log_4{8} ).
• Condensing:
  • Combine logs by dividing: ( \log_6{\frac{9}{3}} ).

Power Rule

• Raising a power inside the log: bring down exponent as multiplication.
  • ( \log_3{8^2} = 2 \log_3{8} ).
  • ( \log_3{x^9} = 9 \log_3{x} ).

Combining Logarithm Properties

• Expansion Example:
  • Multiply terms within to separate with addition.
  • Bring exponents down: ( \log_2{x} + 2 \log_2{y} ).
• Condensing Example:
  • Bring numbers up as exponents.
  • Combine by multiplication/division: ( \log_2{(x^4 \cdot (y + 2)^5 \cdot z^3)} ).

Conclusion

• Logarithms simplify exponent evaluation.
• Understanding properties aids in solving logarithmic equations.
• Practice converting and applying rules for mastery.