Exponentials with Arbitrary Bases Help

Introduction to Exponentials with Arbitrary Bases

We know how to define integer powers of real numbers. For instance

But what does it mean to calculate

4 or π ^e ?

You can calculate values for these numbers by punching suitable buttons on your calculator, but that does not explain what the numbers mean or how the calculator was programmed to calculate them. We will use our understanding of the exponential and logarithm functions to now define these exponential expressions.

If a > 0 and b is any real number then we define

To come to grips with this rather abstract formulation, we begin to examine some properties of this new notion of exponentiation:

If a is a positive number and b is any real number then

(1) ln( a ^b ) = b · ln a .

In fact

ln( a ^b ) = ln(exp( b · ln a )).

But ln and exp are inverse, so that the last expression simplifies to b · ln a .

Examples

You Try It : Simplify the expression ln[ e ^x · x ^e ].

Because of this last example we will not in the future write the exponential function as exp( x ) but will use the more common notation e ^x. Thus

Examples

You Try It : Simplify the expression ln[ e ^{3 x} · e ^{− y −5} · 2 ⁴ ].

Example 5

You Try It : Solve the equation 4 ^x · 3 ^{2 x} = 7. [ Hint : Take the logarithm of both sides. See also Example 6.22 below.]

Find practice problems and solutions for these concepts at: Transcendental Functions Practice Test.