Logarithmic Functions Overview

Overview

This lesson focused on logarithmic functions: their properties, graph transformations, real-world applications, and solving both logarithmic and natural logarithmic equations. The lesson concluded with the use of the change of base formula.

Properties and Applications of Logarithmic Functions

• Logarithmic functions are the inverses of exponential functions.
• The general form is ( f(x) = \log_a(x) ), where ( a ) is the base.
• Common real-world applications include:
  • The Richter scale (measuring earthquake magnitude)
  • The pH scale (measuring acidity)
  • The decibel scale (measuring loudness)
  • Password strength evaluation

Converting Between Logarithmic and Exponential Form

• If no base is shown in "log", the base is 10 (common logarithm).
• To convert ( \log_a(b) = c ) to exponential form: ( a^c = b ).
• Accurate conversion between forms is essential for solving equations.

Solving Logarithmic Equations

• Use logarithmic properties (product, quotient, power rules) to combine or simplify terms before converting to exponential form.
• Isolate the variable by using inverse operations (e.g., divide out coefficients before converting).
• Clearly show each algebraic step for full credit and understanding.

Graphs and Transformations of Logarithmic Functions

• The graph of a logarithmic function has a vertical asymptote at ( x = 0 ).
• Domain: ( (0, \infty) ); Range: ( (-\infty, \infty) ).
• Transformations include:
  • Horizontal and vertical shifts
  • Reflections across axes
  • Stretches and compressions
• Use graphing calculators to check transformations and graph properties.

Applications and Word Problems

• Example: To determine if a solution is acidic, use ( \text{pH} = -\log[H^+] ) and compare the result to 7.
• Enter values in scientific or decimal form as needed; know how to use your calculator’s log functions.

Natural Logarithmic Functions

• The natural logarithm, ( \ln x ), is log base ( e ).
• Key inverse properties:
  • ( \ln(e^x) = x )
  • ( e^{\ln x} = x )
• To solve equations like ( 2e^{3x} = 8 ):
  • Isolate the exponential part
  • Apply ( \ln ) to both sides
  • Simplify and solve for ( x )

Change of Base Formula

• For logarithms with bases other than 10 or ( e ), use:
  • ( \log_a(x) = \frac{\ln x}{\ln a} ) or ( \frac{\log x}{\log a} )
• This allows you to evaluate any logarithm using a calculator.

Practice and Participation Guidance

• Steps for problem-solving: identify given and unknown values, set up the equation, apply properties, solve, and interpret the result.
• Use correct interval notation (parentheses for values not included).
• Always show your work in a clear, logical order to receive full credit.