**Objective: **Define exponential functions that have bases other than e.

Differentiate and integrate exponential functions that have

bases other than e. Use exponential functions to model

compound interest and exponential growth.

Definition of Exponential Function to Base a

If a is a positive real number (a≠1) and x is any real

number, then the exponential function to the base a is

denoted by a^{x} and is defined by

a^{x} = e^{(lna)x}

If a = 1, then y = 1^{x} = 1 is a constant function.

These functions obey the usual laws of exponents

**Definition of Logarithmic Function to Base a**

If a is a positive real number (a≠1) and x is any
positive

real number, then the **logarithmic function to the base
a **is denoted by log

These functions obey the usual laws of logarithms

**Differentiation and Integration
**To differentiate exponential and logarithmic functions to

other bases you have three options

1. use def’n of a

for natural exponential and log functions

2. use logarithmic differentiation

3. use the following differentiation rules for bases other

than e.

**Derivatives for Bases other than e**

Let a be a positive real number (a≠1) and let u be a

differentiable function of x.

Ex: Find the derivative of each function

Ex: Find

Previously the power rule required n to be a rational

number. However, now the rule can be extended to cover

any real value number.

**The Power Rule for Real Exponents**

Let n be any real number and let u be a differentiable

function of x.

b. Differentiate