Exponential and Logarithmic Equations - Study Notes

Key Learning Objectives

  • Exponential Equations

    • Solve using like bases

    • Solve using logarithms

  • Logarithmic Equations

    • Use the definition of a logarithm

    • Apply 1-to-1 properties of logarithms

  • Applications

    • Real-world applications of logarithmic and exponential equations

  • Conversion to Exponential Form

    • Convert the equation ( y = ab^x ) to an exponential equation using base ( e )

Definition of a Logarithm

  • A logarithm is defined as:

    • ( y = ext{log}_b{x} ) is equivalent to ( b^y = x )

    • Inverse Properties:

    • ( b^{ ext{log}_b{x}} = x )

    • ( ext{log}_b{b^x} = x )

Logarithm Properties

  • Special Values:

    • ( ext{log}_b{b} = 1 )

    • ( ext{log}_b{1} = 0 )

  • Product Rule: ( ext{log}b(MN) = ext{log}b{M} + ext{log}_b{N} )

  • Quotient Rule: ( ext{log}b\left(\frac{M}{N}\right) = ext{log}b{M} - ext{log}_b{N} )

  • Power Rule: ( ext{log}b{M^p} = p \cdot ext{log}b{M} )

One-to-One Properties of Logarithms

  • If ( M = N ) then ( ext{log}b{M} = ext{log}b{N} ).

Solving Exponential Equations

  1. General Steps:

    • Isolate the exponential factor.

    • Take logarithm (common or natural) of both sides.

    • Simplify using logarithm properties.

    • Solve for the variable.

    • Verify your solution.

  2. Example:

    • Solve for ( x ): ( 5^x = 60 ):

      • Take ( ext{ln} ) or ( ext{log} ): ( x \cdot ext{ln}(5) = ext{ln}(60) )

      • Thus, ( x = \frac{ ext{ln}(60)}{ ext{ln}(5)} )

Solving Logarithmic Equations

  1. General Steps:

    • Combine logarithms if possible.

    • Convert to exponential form.

    • Solve for the variable.

    • Check your answer.

  2. Example:

    • Solve: ( ext{log}_4(x + 3) = 2 )

      • Convert to exponential: ( x + 3 = 4^2 ) --> ( x = 16 - 3 ) --> ( x = 13 )

Applications of Exponential and Logarithmic Functions

  • Periodic Interest Formula:

    • ( A = P\left(1 + \frac{r}{n}\right)^{nt} )

  • Continuous Interest Formula:

    • ( A = Pe^{rt} )

  • Example Problems:

    • Time to grow $25,000 to $500,000 at different rates.

Exponential Growth and Decay

  • Growth Model:

    • ( A(t) = A_0 e^{kt} )

  • Decay Model:

    • Understanding half-lives and decay rates. Example: Carbon-14 decay.

Linear vs Exponential Functions

  • Recognizing and distinguishing between linear and exponential behavior based on data tables or equations.

  • Examples showing the identification of functions as linear or exponential and formulating equations based on their behavior.

Homework Problems & Practice Examples

  • Solve exponential and logarithmic equations through various methods including factoring and properties of logarithms.