Lecture on Logarithms

Overview

• Focus on logarithms: evaluating logs, change of base formula, expanding/condensing logs, solving equations, graphing logarithmic functions.

Evaluating Logs

• Logarithmic Evaluation: To find ( \log_{b}(n) ), ask how many times ( b ) must be multiplied by itself to reach ( n ).
  • Example: ( \log_{2}(4) = 2 ) because ( 2^2 = 4 ).
  • Other Examples:
    • ( \log_{2}(8) = 3 ) (( 2^3 = 8 ))
    • ( \log_{3}(9) = 2 ) (( 3^2 = 9 ))
    • ( \log_{4}(16) = 2 ), ( \log_{3}(27) = 3 ), ( \log_{2}(32) = 5 )._

Change of Base Formula

• Formula: ( \log_{a}(b) = \frac{\log_{c}(b)}{\log_{c}(a)} ) - allows changing the base to any number, usually 10 or ( e ).
  • Example: ( \log_{4}(16) = 2 ) is verifiable by ( \log(16)/\log(4) = 2 ).

Properties of Logs

• Addition: ( \log(a) + \log(b) = \log(ab) ).
• Subtraction: ( \log(a) - \log(b) = \log(\frac{a}{b}) ).
• Exponentiation: ( \log(a^b) = b \log(a) ).

Expanding and Condensing Logs

• Expanding involves breaking down log expressions into sums/differences.

  • Use exponent rules to aid expansion.
  • Example: ( \log(x^2y/z) ) becomes ( 2\log(x) + \log(y) - \log(z) ).

• Condensing involves the reverse, combining log terms into a single log expression.

Solving Logarithmic Equations

• Convert the log equation into exponential form to solve.

  • Example: ( \log_{2}(x) = 3 ) is ( 2^3 = x ), so ( x = 8 ).

• Extraneous Solutions: Check for solutions that don't satisfy the domain of logarithmic functions (e.g., logs of negative numbers)._

Graphing Logarithmic Functions

• Vertical Asymptotes: Occur where the inside expression of the log is zero.

• Domain and Range:

  • Domain: Values ( x > 0 ) for ( \log(x) ).
  • Range: All real numbers.

• Inverse Functions: Logarithmic and exponential functions are inverses.

Examples

• Logarithmic to Exponential:

  • ( \log_{3}(27) = 3 ) becomes ( 3^3 = 27 ).

• Exponential to Logarithmic:

  • ( 2^3 = 8 ) becomes ( \log_{2}(8) = 3 ).

Special Logarithms

• Natural Logs: ( \ln(x) ) with base ( e ).

• Typical Values:

  • ( \ln(1) = 0 ), ( \ln(e) = 1 ).

This summary covers the key elements discussed in the lecture on logarithms. It includes methods for manipulating and solving log expressions and equations, properties of logarithms, and notes on graphing and changing the base of logarithmic expressions.