Lecture Notes: Exponential and Logarithmic Functions

Exponential Functions

• Definition: A function where the variable x is the exponent.
• Example: ( y = e^x )
  • ( e ) (Euler's number) ( \approx 2.7 ).
• Graph Characteristics:
  • Passes through the point (0,1) because ( e^0 = 1 ).
  • For negative x-values, ( e^x ) approaches 0 but never reaches it.
  • For increasing x-values, ( e^x ) grows rapidly (exponential growth).
• Real-World Example: Growth of users on a popular website.

Logarithmic Functions

• Definition: The inverse of an exponential function.
• Example: ( y = \ln(x) ), the natural logarithm, equivalent to ( \log_e(x) ).
• Graph Characteristics:
  • Passes through the point (1,0) because ( \ln(1) = 0 ).
  • Similar shape to exponential functions but reflected over the line ( y = x ).
  • The natural log function is the inverse of the exponential function.

Relationship Between Exponentials and Logarithms

• Inverse Functions: ( y = e^x ) and ( y = \ln(x) ) are inverses, reflections across ( y = x ).

Transformations of Functions

• Functions can be transformed using shifts and stretches.
• Example: ( y = 2e^{x+1} - 5 )
  • Vertical Shift: (-5), down 5 units.
  • Horizontal Shift: (+1), left 1 unit.
  • Vertical Stretch: (2), stretches vertically by a factor of 2.

Key Points for IB Maths

• Understand the basic shapes of exponential and logarithmic functions.
• Function transformations involve basic shifts and stretches.
• Use calculators for sketching functions and finding intercepts in exams.
• Exponential decay occurs when there's a negative sign in the exponent.
• Flipping occurs with a negative sign in front of the whole function.

Conclusion

• Exponential and logarithmic functions are closely related as inverses.
• Mastering transformations and graph characteristics is important for solving IB math problems related to these functions.