Exponential and Logarithmic Functions

Before their derivatives can be found (Section 7), the exponential and logarithmic functions

themselves need a precise definition and a summary of their key properties.

The exponential function. For a fixed base $a>0$ ($a\neq1$), the function $y=a^x$ is defined

for every real $x$, with $a^x>0$ always. If $a>1$, $a^x$ is strictly increasing; if $0<a<1$, it is

strictly decreasing. In every case its graph passes through $(0,1)$ (since $a^0=1$) and never

touches the $x$-axis, approaching it asymptotically as $x\to-\infty$ (for $a>1$).

The number $e$. Among all possible bases, one particular value -- denoted $e$, and equal to

$2.71828\ldots$ (irrational) -- is singled out as the natural base, because it is precisely

the base for which the exponential function's own derivative equals itself (established in

Section 7). It arises as the limit $e=\lim_{n\to\infty}(1+1/n)^n$, though this limit is stated

here rather than evaluated. The function $y=e^x$, the natural exponential function, is the

single most important exponential function in calculus, appearing throughout differential

equations, compound growth and decay, and probability.

The natural logarithm. Since $y=e^x$ is one-one and increasing on all of $\mathbb{R}$, it has

an inverse function, defined for $x>0$ and denoted $\ln x$ (or $\log_e x$): by definition,

Because $\ln x$ is the inverse of $e^x$, their graphs are mirror images of each other in the line

$y=x$; in particular $\ln x\to-\infty$ as $x\to0^+$, $\ln1=0$, and $\ln x$ increases (slowly,

without bound) as $x\to\infty$.

Key algebraic properties of $\ln$, all following from the corresponding index laws for $e^x$:

These four laws -- turning a product into a sum, a quotient into a difference, and

a power into a product -- are exactly what makes logarithmic differentiation (Section 8)

so effective on products, quotients and variable powers.

General base logarithm. For any base $a>0,\ a\neq1$, $\log_a x$ is defined as the inverse of …