Properties of the exponential functions

Operations for functions: Exponential and logarithmic functions

Properties of the exponential functions

Properties of the exponents

Let $a$ and $b$ be two positive real numbers. The following rules of calculation hold for the exponential functions $a^x$ and $b^x$ .

1	$a^0=1$
2	$a^{x+y}=a^x\cdot a^y$
3	$a^{-x}=\frac{1}{a^x}=\left (\frac{1}{a} \right )^x$
4	$a^{x-y}=\frac{a^x}{a^y}$
5	$\left(a^x\right)^y = a^{x\cdot y}$
6	$(a\cdot b)^x = a^x\cdot b^x$
7	if $a\lt b$ and $x\gt 0$ , then $a^x\lt b^x$
8	if $a\gt 1$ and $x\lt y$ , then $a^x\lt a^y$

In particular, $a^x$ is increasing if $a\gt1$ and decreasing if $a\lt 1$ .

These rules are direct consequences of the known calculation rules for arithmetic.

By way of example, we use the rules to show why $a^x$ is decreasing if $a\lt 1$ . Suppose $x$ and $y$ are real numbers with $x\lt y$ . Then we have:

$\begin{array}{rcl}\left(\frac{1}{a}\right)^x&\lt& \left(\frac{1}{a}\right)^y\\ &&\phantom{xx}\color{blue}{\text{rule 8}}\\ \left(\frac{1}{a}\right)^x\cdot a^{x+y}&\lt& \left(\frac{1}{a}\right)^y\cdot a^{x+y}\\ &&\phantom{xx}\color{blue}{\text{both sides multiplied by the same positive number}}\\ {a}^{-x}\cdot a^{x+y}&\lt& {a}^{-y}\cdot a^{x+y}\\ &&\phantom{xx}\color{blue}{\text{rule 3}}\\ a^{y}&\lt& a^{x}\\ &&\phantom{xx}\color{blue}{\text{rule 2}} \end{array}$ This shows that $a^x\gt a^y$ , so $a^x$ is decreasing.

Here are some examples:

1	$2^0=1$
2	$3^{x+y}=3^3\cdot 3^y$
3	$5^{-x}=\frac{1}{5^x}=\left (\frac{1}{5} \right )^x$
4	$4^{2}=\frac{4^5}{4^3}$
5	$\left(2^5\right)^6 = 2^{30}$
6	$6^x = 2^x\cdot 3^x$

Simplify the expression $a^7\cdot a^8$ .

$a^7\cdot a^8=$ $a^{15}$

From $a^x\cdot a^y=a^{x+y}$ we deduce $a^7\cdot a^8=a^{7+8}=a^{15}$ .

New example