Integrals, Exponential Functions, and Logarithms: Learn It 3

General Logarithmic and Exponential Functions

We close this section by looking at exponential functions and logarithms with bases other than [latex]e.[/latex]

Exponential functions are functions of the form [latex]f(x)={a}^{x}.[/latex] Note that unless [latex]a=e,[/latex] we still do not have a mathematically rigorous definition of these functions for irrational exponents.

Let’s rectify that here by defining the function [latex]f(x)={a}^{x}[/latex] in terms of the exponential function [latex]{e}^{x}.[/latex] 

definition of general exponential functions

For any [latex]a>0,[/latex] and for any real number [latex]x,[/latex] define [latex]y={a}^{x}[/latex] as follows:

[latex]y={a}^{x}={e}^{x\text{ln }a}[/latex]

Now [latex]{a}^{x}[/latex] is defined rigorously for all values of [latex]x[/latex].

This definition also allows us to generalize property iv. of logarithms and property iii. of exponential functions to apply to both rational and irrational values of [latex]r.[/latex] It is straightforward to show that properties of exponents hold for general exponential functions defined in this way.

Let’s now apply this definition to calculate a differentiation formula for [latex]{a}^{x}.[/latex] We have

[latex]\frac{d}{dx}{a}^{x}=\frac{d}{dx}{e}^{x\text{ln }a}={e}^{x\text{ln }a}\text{ln}a={a}^{x}\text{ln }a.[/latex]

The corresponding integration formula follows immediately.

derivatives and integrals involving general exponential functions

Let [latex]a>0.[/latex] Then,

[latex]\frac{d}{dx}{a}^{x}={a}^{x}\text{ln }a[/latex]

and

[latex]\displaystyle\int {a}^{x}dx=\frac{1}{\text{ln }a}{a}^{x}+C[/latex]

If [latex]a\ne 1,[/latex] then the function [latex]{a}^{x}[/latex] is one-to-one and has a well-defined inverse. Its inverse is denoted by [latex]{\text{log}}_{a}x.[/latex] Then,

[latex]y={\text{log}}_{a}x\text{if and only if}x={a}^{y}[/latex]

Note that general logarithm functions can be written in terms of the natural logarithm. Let [latex]y={\text{log}}_{a}x.[/latex] Then, [latex]x={a}^{y}.[/latex] Taking the natural logarithm of both sides of this second equation, we get

[latex]\begin{array}{ccc}\hfill \text{ln}x& =\hfill & \text{ln}({a}^{y})\hfill \\ \hfill \text{ln}x& =\hfill & y\text{ln}a\hfill \\ \hfill y& =\hfill & \frac{\text{ln}x}{\text{ln}a}\hfill \\ \hfill {\text{log}}_{}x& =\hfill & \frac{\text{ln}x}{\text{ln}a}.\hfill \end{array}[/latex]

Thus, we see that all logarithmic functions are constant multiples of one another. Next, we use this formula to find a differentiation formula for a logarithm with base [latex]a.[/latex] Again, let [latex]y={\text{log}}_{a}x.[/latex] Then,

[latex]\begin{array}{cc}\hfill \frac{dy}{dx}& =\frac{d}{dx}({\text{log}}_{a}x)\hfill \\ & =\frac{d}{dx}(\frac{\text{ln}x}{\text{ln}a})\hfill \\ & =(\frac{1}{\text{ln}a})\frac{d}{dx}(\text{ln}x)\hfill \\ & =\frac{1}{\text{ln}a}·\frac{1}{x}\hfill \\ & =\frac{1}{x\text{ln}a}.\hfill \end{array}[/latex]

derivatives of general logarithm functions

Let [latex]a>0.[/latex] Then,

[latex]\frac{d}{dx}{\text{log}}_{a}x=\frac{1}{x\text{ln}a}[/latex]

Evaluate the following derivatives:

  1. [latex]\frac{d}{dt}({4}^{t}·{2}^{{t}^{2}})[/latex]
  2. [latex]\frac{d}{dx}{\text{log}}_{8}(7{x}^{2}+4)[/latex]

Evaluate the following integral: [latex]\displaystyle\int {x}^{2}{2}^{{x}^{3}}dx.[/latex]