14.4 The Exponential Function

14.29   Example. We will now derive some properties of the inverse function $E$ of the logarithm.

We have

$$\begin{eqnarray*} \ln(1) = 0 & \mbox{$\Longrightarrow$}& E(0) = 1, \\ \ln(e) = 1 & \mbox{$\Longrightarrow$}& E(1) = e. \end{eqnarray*}$$

For all $a$ and $b$ in R,

$$a+b = \ln(E(a)) + \ln(E(b)) = \ln( E(a)E(b) ).$$

If we apply $E$ to both sides of this equality we get

$$E(a+b) = E(a)E(b) \mbox{ for all }a,b\in \mbox{{\bf R}}.$$

For all $a\in\mbox{{\bf R}}$ we have

$$1 = E(0) = E(a+(-a)) = E(a)E(-a),$$

from which it follows that

$$E(-a) = (E(a))^{-1} \mbox{ for all }a \in \mbox{{\bf R}}.$$

If $a\in\mbox{{\bf R}}$ and $q \in \mbox{{\bf Q}}$ we have

$$\ln( (E(a))^q) = q\ln(E(a)) = qa.$$

If we apply $E$ to both sides of this identity we get

$$(E(a))^q = E(qa) \mbox{ for all }a\in\mbox{${\mbox{{\bf R}}}^{+}$}, q\in \mbox{{\bf Q}}.$$

In particular,

$$e^q = (E(1))^q = E(q) \mbox{ for all }q\in\mbox{{\bf Q}}.$$ (14.30)

Now we have defined $E(x)$ for all $x \in \mbox{{\bf R}}$, but we have only defined $x^q$ when $x \in \mbox{${\mbox{{\bf R}}}^{+}$}$ and $q \in \mbox{{\bf Q}}$. (We know what $${ 2^{1 \over 2}}$$ is, but we have not defined $2^{\sqrt 2}$.) Because of relation (14.30) we often write $e^x$ in place of $E(x)$. $E$ is called the exponential function, and is written

$$E(x) = e^x = \exp(x) \mbox{ for all }x \in \mbox{{\bf R}}.$$

We can summarize the results of this example in the following theorem:

14.31   Theorem (Properties of the exponential function.) The exponential function is a function from R onto $\mbox{${\mbox{{\bf R}}}^{+}$}$. We have

$$e^{a+b} = e^a e^b \mbox{ for all }a,b \in \mbox{{\bf R}}.\mbox{{}}$$

$$e^{a-b} = {e^a \over e^b} \mbox{ for all }a,b \in \mbox{{\bf R}}.$$ (14.32)

$$(e^a)^q = e^{aq} \mbox{ for all }a\in\mbox{{\bf R}},\mbox{ and }\mbox{ for all }q \in\mbox{{\bf Q}}.\mbox{{}}$$

$$(e^a)^{-1} = e^{-a} \mbox{ for all }a \in\mbox{{\bf R}}.\mbox{{}}$$

$$e^{\ln(x)} = x \mbox{ for all }x \in\mbox{${\mbox{{\bf R}}}^{+}$}.\mbox{{}}$$

$$\ln(e^a) = a \mbox{ for all }a \in\mbox{{\bf R}}.\mbox{{}}$$

$$e^0 = 1.\mbox{{}}$$

$$e^1 = e.$$ (14.33)

Proof: We have proved all of these properties except for relation (14.32). The proof of (14.32) is the next exercise.

14.34   Exercise. Show that $${e^{a-b} = {e^a \over e^b} \mbox{ for all }a,b \in \mbox{{\bf R}}}$$.

14.35   Exercise. Show that if $a \in \mbox{${\mbox{{\bf R}}}^{+}$}$ and $q \in \mbox{{\bf Q}}$, then

$$a^q = e^{q\ln(a)}.$$

14.36   Definition ($a^x$.) The result of the last exercise motivates us to make the definition

$$a^x = e^{x\ln(a)} \mbox{ for all }x\in\mbox{{\bf R}}\mbox{ and }\mbox{ for all }a\in\mbox{${\mbox{{\bf R}}}^{+}$}.$$

14.37   Exercise. Prove the following results:

$$\begin{eqnarray*} a^x a^y &=& a^{x+y} \mbox{ for all }a\in\mbox{${\mbox{{\bf R}}... ...x{{\bf R}}}^{+}$}\mbox{ and }\mbox{ for all }x\in\mbox{{\bf R}}. \end{eqnarray*}$$