Logarithmic and Exponential Functions with Other Bases

General Exponential Functions

Exponential functions of the form f(x) = bx are fundamental in calculus, especially when the base b is not equal to the natural base e. The properties of these functions depend on the value of b:

• Domain:

• Range:

• For all :

• If : is an increasing function

• If : is a decreasing function

Exponential functions can be rewritten using the natural exponential function:

This representation is useful for differentiation and integration.

General Logarithmic Functions

The logarithmic function with base b, denoted logbx, is the inverse of the exponential function bx. For any base , , the function is one-to-one and has the following properties:

• Domain:

• Range:

• logb1 = 0 for any base ,

• If : is an increasing function

Inverse Relations for Exponential and Logarithmic Functions

Exponential and logarithmic functions are inverses of each other. The following relations hold for any base , :

• , for

• , for all

Derivatives of Exponential Functions

The derivative of an exponential function with base is given by:

• For a function ,

Example: Find the derivative of :

Integrals of Exponential Functions

The indefinite integral of an exponential function with base is:

Example:

The General Power Rule

The general power rule applies to functions of the form :

• , for real and

• If is a positive differentiable function,

Example:

Derivatives of General Logarithmic Functions

The derivative of the logarithmic function with base is:

• , for

• For ,

Summary of New Rules

Additional info:

Some images, such as image_6, depict trigonometric unit circle values and are not directly relevant to the topic of logarithmic and exponential functions, so they are not included.