Step By Step Calculus » 7.1 - Exponential Functions

show_hiddenExpand All [+]
Exponential functions are functions of the form f(x)=a^xf(x)=a^x. The fixed number aa is the base, and the variable xx is the exponent. These functions have enormous importance in pure and applied mathematics.
If a>0a>0, then a^{x}a^{x} can be defined for all x\in\mathbb{R}x\in\mathbb{R}, but if a<0a<0 this is not possible. For x = n \in \mathbb{N}, \; \; a^{n} x = n \in \mathbb{N}, \; \; a^{n} is just aa multiplied by itself nn times---i.e., aa to the n^{th}n^{th} power. For a negative natural number xx, say x = -n, \; \; a^{-n} = \dfrac{1}{a^{n}}x = -n, \; \; a^{-n} = \dfrac{1}{a^{n}}.
For a > 0a > 0 and rational xx i.e. for x=\frac{m}{n}, \; \; m \in \mathbb{Z}, \; n \in \mathbb{N}, x=\frac{m}{n}, \; \; m \in \mathbb{Z}, \; n \in \mathbb{N},
\displaystyle a^{\frac{m}{n}}=\left(a^{\frac{1}{n}}\right)^m=(a^m)^{\frac{1}{n}}
\displaystyle a^{\frac{m}{n}}=\left(a^{\frac{1}{n}}\right)^m=(a^m)^{\frac{1}{n}}
where the n^{th}n^{th} root is defined to be the positiven^{th}n^{th} root of aa.
If a<0a<0, consider a^{\frac{m}{n}}a^{\frac{m}{n}} with the fraction \dfrac{m}{n}\dfrac{m}{n} in lowest terms (i.e., mm and nn have no factors in common). Then this expression makes sense if and only if nn is odd (in which case mm can be even or odd). For this reason we are going to focus on a^{x}a^{x} with a > 0a > 0 and will restrict ourselves to the occasional comment on the case when a < 0a < 0.
By definition\;a^{0} = 1\;a^{0} = 1 for any a \neq 0a \neq 0, and 0^{0}0^{0} is undefined. The algebra of exponential functions is governed by the properties of the exponential, which for a > 0, \; \; b > 0a > 0, \; \; b > 0 and x, \, y \in \mathbb{ R}x, \, y \in \mathbb{ R}, are:
  • a^{-x} = \dfrac{1}{a^x}a^{-x} = \dfrac{1}{a^x}
  • a^{x+y}=a^{x}a^{y}a^{x+y}=a^{x}a^{y}
  • (a^{x})^{y}=a^{xy}(a^{x})^{y}=a^{xy}
  • (ab)^{x}=a^{x}b^{x}(ab)^{x}=a^{x}b^{x}
  • If a>1a>1, then a^{x}<a^{y}a^{x}<a^{y} if and only if x<yx<y
One interesting result which follows from these properties is
\displaystyle \vert a\cdot b\vert=\vert a\vert \cdot \vert b\vert \; \forall a, b \in \mathbb{R}.
\displaystyle \vert a\cdot b\vert=\vert a\vert \cdot \vert b\vert \; \forall a, b \in \mathbb{R}.
The value of aa for which the graph of a^xa^x goes through the point (0,1)(0,1) with slope 11 is called the natural base and denoted by the symbol ee; the decimal expansion of the irrational number ee begins 2.71828 \ldots \; 2.71828 \ldots \; . The numbers \pi\pi and ee are perhaps the most famous irrational numbers in mathematics. The corresponding exponential function e^xe^x is called the natural exponent function, or more commonly “the exponential function". An alternative way of writing e^xe^x is \exp(x)\exp(x).