Natural Logarithms Lecture Notes

Introduction to Natural Logarithms

• Natural Logarithm (ln): A logarithm with base ( e )
  • Special notation: ln
• Constant ( e ):
  • Similar to ( \pi )
  • Value of ( e ) used in calculations

Examples and Calculations

• Using ( e ):
  • ( e^{0.5} \approx 1.6487 )
  • ( e^{-2.67} \approx 0.0693 )
• Using Natural Log (ln):
  • ( \ln(3) \approx 1.0986 )
  • ( \ln\left(\frac{1}{4}\right) \approx -1.3863 )

Switching Between Exponential and Logarithmic Forms

• Exponential Form:
  • Example: ( x ) as an exponent.
  • Log base ( e ) of 23: ( \ln(23) = x )
• Logarithmic Form:
  • Example: ( \ln(x) \approx 1.2528 )
  • Convert to exponential: ( e^{1.2528} = x )

Inverse Property

• Inverse Property:
  • Base ( e ) and natural logarithms cancel out.
  • Example: ( e^{ln(x)} = x )
• Examples:
  • ( e^{\ln(21)} = 21 )
  • ( \ln(e^{x^2 - 1}) = x^2 - 1 )

Solving Equations

Example 1

• Equation: ( \ln(3x) = 0.5 )
  • Convert to exponential: ( e^{0.5} = 3x )
  • Calculate: ( 1.6487 = 3x )
  • Solve for ( x ): ( x = 0.5496 )

Example 2

• Equation: ( \ln(2x - 3) = 2.5 )
  • Convert to exponential: ( e^{2.5} = 2x - 3 )
  • Calculate: ( 12.1825 = 2x - 3 )
  • Solve for ( x ): ( x = 7.5912 )

Solving Exponential Equations

Example 3

• Equation: ( 3e^{-2x} = 10 )
  • Isolate exponential: ( 3e^{-2x} = 6 )
  • Convert using ( \ln ): ( \ln(e^{-2x}) = \ln(2) )
  • Inverse property: ( -2x = \ln(2) )
  • Solve for ( x ): ( x = -0.3466 )

Example 4

• Equation: ( 2e^{x+5} = 14 )
  • Isolate exponential: ( e^{x+5} = 7 )
  • Convert using ( \ln ): ( \ln(e^{x+5}) = \ln(7) )
  • Inverse property: ( x + 5 = 1.9459 )
  • Solve for ( x ): ( x = -3.0541 )

Conclusion

• Reviewed natural logarithms and properties
• Practiced converting between forms and solving equations
• Emphasized the importance of inverse properties in solving equations