Graphing Natural Log and Exponential Functions with Base e

Key Concepts

• Graphing natural log and exponential functions with base e is similar to graphing regular logarithmic and exponential functions.
• Natural Log (ln): Natural logarithm functions have a base of e.
• Exponential Functions: Functions expressed as ( e^x ).

Graphing ( e^{x-2} + 3 )

• Steps:
  1. Set the exponent equal to 0 and 1 to find key points. Solve for x.
  2. Use x-values from solving the equation (2 and 3 in this example).
  3. Identify the horizontal asymptote from the equation: ( y = 3 ).
• Y Values:
  • For ( x = 2 ):
    • ( e^{0} = 1 )
    • ( 1 + 3 = 4 )
  • For ( x = 3 ):
    • ( e^{1} \approx 2.72 )
    • ( 2.72 + 3 \approx 5.72 )
• Graph Details:
  • Horizontal asymptote at ( y = 3 ).
  • Points: ( (2, 4) ) and ( (3, 5.7) ).
  • Domain: All real numbers ((-\infty, \infty)).
  • Range: (3, \infty)).

Graphing ( \ln(x-1) + 2 )

• Steps:
  1. Set the inside equal to 0, 1, and e. Solve for x.
  2. This gives key points (1, 2, and 3.7).
  3. Identify the vertical asymptote from the equation: ( x = 1 ).
• Y Values:
  • For ( x = 2 ):
    • ( \ln(1) = 0 )
    • ( 0 + 2 = 2 )
  • For ( x = 3.7 ):
    • ( \ln(e) = 1 )
    • ( 1 + 2 = 3 )
• Graph Details:
  • Vertical asymptote at ( x = 1 ).
  • Points: ( (2, 2) ) and ( (3.7, 3) ).
  • Domain: ([1, \infty)).
  • Range: All real numbers ((-\infty, \infty)).](streamdown:incomplete-link)

Summary

• The techniques for graphing natural log functions and exponential functions with base e are consistent with those for regular logarithmic and exponential functions.
• Focus on recognizing asymptotes and plotting key points derived from solving equations related to the function.