2.4.1 Exponential and Logarithmic Functions

Exponential Function Definition

An exponential function is defined by f(x) = a^x, a > 0

Special Exponential Function

• An important exponential function is 

  • Where ‘e’ is the mathematical constant 2.718…

• An exponential function can be written using ‘e’: a^x = e^{x lna}

Exponential Function: Domain

Its domain is the set of all real values

Exponential Function: Range

Its range is the set of all positive real values

Key Features of Exponential Graphs

Logarithmic Function Definition

A logarithmic function is of the form f(x) = log_ax, x > 0

Logarithmic Function: Domain

Its domain is the set of all positive real values

• You can't take a log of zero or a negative number

Logarithmic Function: Range

Its range is set of all real values

Relationship Between Exponential and Logarithmic Functions

• ‘y = log_ax’ and' ‘y = a^x’ are inverse functions

• No need to first switch the variables and solve the equation to get the inverse like with most other functions. Just immediately write this translation

Natural Logarithmic Functions

An important logarithmic function is f(x) = lnx

• lnx = log_ex

Relationship Between Special Exponential and Natural Logarithmic Functions

• ‘y = ln x’ and' ‘y = e^x’ are inverse functions

  • Because ln e^x = x and e^{ln x} = x

• No need to first switch the variables and solve the equation to get the inverse like with most other functions. Just immediately write this translation

Relationship Between Logarithmic and Natural Logarithmic Functions

• Any logarithmic function can be written using ln

log_ax = \frac{ln x}{ln a}, using the change of base formula

Key Features of Logarithmic Graphs