4.4 The exponential and logarithm functions

We derive next the elementary properties of the exponential and logarithmic functions. Of course, by “exponential function,” we mean the power series function $exp.$ And, as yet, we have not even defined a logarithm function.

Using part (d) of the preceding exercise, we make the following important definition.

We call the inverse ${exp}^{-1}$ of the restriction of the exponential function to $R$ the (natural) logarithm function, and we denote this function by $ln.$

The properties of the exponential and logarithm functions are strongly tied to the simplestkinds of differential equations. The connection is suggested by the fact, we have already observed, that ${exp}^{\text{'}}=exp.$ The next theorem, corollary, and exercises make these remarks more precise.

Suppose $f:C\to C$ is differentiable everywhere and satisfies thedifferential equation $f^{\text{'}}=af,$ where $a$ is a complex number. Then $f\left(z\right)=cexp\left(az\right),$ where $c=f\left(0\right).$

Consider the function $h\left(z\right)=f\left(z\right)/exp\left(az\right).$ Using the Quotient Formula, we have that

Hence, there exists a complex number $c$ such that $h\left(z\right)=c$ for all $z.$ Therefore, $f\left(z\right)=cexp\left(az\right)$ for all $z.$ Setting $z=0$ gives $f\left(0\right)=c,$ as desired.

Law of exponents

For all complex numbers $z$ and $w,$ $exp(z+w)=exp\left(z\right)exp\left(w\right).$

Fix $w,$ define $f\left(z\right)=exp(z+w),$ and apply the preceding theorem. We have $f^{\text{'}}\left(z\right)=exp(z+w)=f\left(z\right),$ so we get

We are about to make the connection between the number $e$ and the exponential function. The next theorem is the first step.

$ln\left(1\right)=0$ and $ln\left(e\right)=1.$

If we write $1=exp\left(t\right),$ then $t=ln\left(1\right).$ But $exp\left(0\right)=1,$ so that $ln\left(1\right)=0,$ which establishes the first assertion.

Recall that

Therefore,

This establishes the second assertion of the theorem.

We have finally reached a point in our development where we can make sense of raising any positive number to an arbitrary complex exponent.Of course this includes raising positive numbers to irrational powers. We make our general definition based on part (c) of the preceding exercise.

For $a$ a positive real number and $z$ an arbitrary complex number, define $a^z$ by

$$a^z=exp\left(zln\left(a\right)\right).$$

REMARK The point is that our old understanding of what $a^r$ means, where $a>0$ and $r$ is a rational number, coincides with the function $exp(rln(a\left)\right).$ So, this new definition of $a^z$ coincides and is consistent with our old definition. And, it now allows us to raies a positive number $a$ to an arbitrary complex exponent.

REMARK Let the bugles sound!! Now, having made all the appropriate definitions and derived all therelevant theorems, we can finally prove that $e^{i\pi }=-1.$ From the definition above, we see that if $a=e,$ then we have $e^z=exp\left(z\right).$ Then, from part (c) of [link] , we have what we want: