Lecture on Logarithms

Introduction to Logarithms

• Logarithms are the inverses of exponential functions.
  • Similar to how square roots are the inverse of squaring.
  • Used to undo what exponential functions do.

Converting Between Logarithmic and Exponential Forms

• Logarithmic form: ( \log_b x = n )
• Exponential form: ( x = b^n )
• Conversion method:
  • Exponentiate both sides using the same base to switch forms.
  • Log base and exponential base cancel each other.
• Base, Exponent, Argument:
  • Base ( b ), Exponent ( n ), Argument/Answer ( x )
  • Example: ( \log_3 81 = 4 ) converts to ( 3^4 = 81 )

Evaluating Logarithms

• Evaluating means finding the value.
• Technique:
  • Set the log equal to a variable.
  • Exponentiate both sides.
• Examples:
  • ( \log_2 32 = x ) where ( 2^5 = 32 ).
  • ( \log_5 125 = x ) where ( 5^3 = 125 ).

Graphing Logarithmic Functions

• Exponential Graphs: Growth function.
• Logarithmic Graphs: Inverse of exponential graphs.
  • Reflected over the line ( y = x ).
  • Vertical asymptote at y-axis.
• Graphing Example:
  • ( y = \log_4 x ): Convert to exponential form and plot.

Properties of Logs

• Product Property: ( \log_b (MN) = \log_b M + \log_b N )
• Quotient Property: ( \log_b \left( \frac{M}{N} \right) = \log_b M - \log_b N )
• Power Property: ( \log_b (M^p) = p \log_b M )
• Change of Base Formula: ( \log_b C = \frac{\log_a C}{\log_a B} )

Expanding and Condensing Logs

• Expanding: Writing a single log as a sum/difference.
  • Use product and quotient properties.
• Condensing: Combining multiple logs into one.
  • Reverse of expanding.
  • Bring exponents down as coefficients.

Solving Logarithmic Equations

• Steps:
  • Isolate the log term.
  • Exponentiate to remove the log.
• Example:
  • Solve ( \log_2 x = 7 ) by converting to ( 2^7 = x ).
• Check for extraneous solutions (cannot log zero/negative).

Conclusion

• Logarithms are critical for solving exponential problems.
• Understanding properties and conversions is key.
• For more help, check out further resources and courses.