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 e.

Exponential functions are functions of the form f(x)=ax. Note that unless a=e, we still do not have a mathematically rigorous definition of these functions for irrational exponents.

Let’s rectify that here by defining the function f(x)=ax in terms of the exponential function ex. 

definition of general exponential functions

For any a>0, and for any real number x, define y=ax as follows:

y=ax=exln a

Now ax is defined rigorously for all values of x.

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 r. 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 ax. We have

ddxax=ddxexln a=exln alna=axln a.

The corresponding integration formula follows immediately.

derivatives and integrals involving general exponential functions

Let a>0. Then,

ddxax=axln a

and

axdx=1ln aax+C

If a1, then the function ax is one-to-one and has a well-defined inverse. Its inverse is denoted by logax. Then,

y=logaxif and only ifx=ay

Note that general logarithm functions can be written in terms of the natural logarithm. Let y=logax. Then, x=ay. Taking the natural logarithm of both sides of this second equation, we get

lnx=ln(ay)lnx=ylnay=lnxlnalogx=lnxlna.

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 a. Again, let y=logax. Then,

dydx=ddx(logax)=ddx(lnxlna)=(1lna)ddx(lnx)=1lna·1x=1xlna.

derivatives of general logarithm functions

Let a>0. Then,

ddxlogax=1xlna

Evaluate the following derivatives:

  1. ddt(4t·2t2)
  2. ddxlog8(7x2+4)

Evaluate the following integral: x22x3dx.