Understanding Logarithms

What are Logarithms?

• Logarithms are the inverse of exponential functions.
• Similar to how square roots undo squaring, logarithms undo exponentiation.
• Useful for solving equations involving exponents.

Switching Between Logarithmic and Exponential Forms

• Logarithmic Form: ( \log_b{x} = n )
• Exponential Form: ( x = b^n )
• To switch, exponentiate both sides of the equation using the same base.
  • Example: ( \log_3{81} = 4 ) translates to ( 3^4 = 81 ).
• Understanding this switch is crucial for solving logarithmic problems.

Graphing Logarithms

• Exponential functions grow, while logarithmic functions are their inverses.
• Graphs of logarithms are reflections of exponential graphs over the line ( y = x ).
• Asymptotes:
  • Logarithmic graphs approach the y-axis (vertical asymptote).
  • Exponential graphs approach the x-axis (horizontal asymptote).
• Example: Graphing ( y = \log_4{x} )
  • Use exponential form ( x = 4^y ) to find points and plot.

Properties of Logarithms

• Product Rule: ( \log_b{(xy)} = \log_b{x} + \log_b{y} )
• Quotient Rule: ( \log_b{\left(\frac{x}{y}\right)} = \log_b{x} - \log_b{y} )
• Power Rule: ( \log_b{x^n} = n \cdot \log_b{x} )
• Change of Base Formula: ( \log_b{x} = \frac{\log_a{x}}{\log_a{b}} )

Expanding and Condensing Logs

• Expanding: Breaking down a log expression into a sum or difference.
  • Example: ( \log{(a \cdot b)} = \log{a} + \log{b} )
• Condensing: Combining multiple logs into a single log expression.
  • Example: ( \log{x} + \log{y} = \log{(xy)} )

Solving Equations with Logarithms

• Isolating Variables: Work from the outside in to isolate the variable.
• Use properties of logarithms to simplify and solve.
• Example Problems:
  • Solve ( 2^{3x} = \frac{3}{4} ) by taking log base 2.
  • Solve ( \log_2{x} = 7 ) by exponentiating with base 2.

Evaluating Logs

• Find the value of logarithmic expressions using properties.
• Using logs to reverse engineer exponential functions.
• Common Logs and Natural Logs:
  • ( \log{x} ) typically implies base 10 (common log).
  • ( \ln{x} ) implies base ( e ) (natural log).

Additional Tips

• Practice by switching between forms and solving various problems to build comfort.
• Watch out for extraneous solutions, especially when logs involve negative values or zero.

Resources

• Check out video courses on ACT/SAT math preparation.
• Further learning available on Mario's Math Tutoring channel.