Powers, roots, and logarithms

Multiply a positive number by itself and the exponent counts the factors. Multiply those two results and the exponents add. Counting factors is enough while the exponent is a positive integer. It does not say what $a^{\pi }$ is, and it does not name the exponent that produces a given height.

Heights multiply when exponents add

Let $a>0$. The curve $y=a^x$ already shows the arithmetic: the height at one exponent, times the height at another, is the height at the sum.

For every positive $a$ and $b$, and every real $x$ and $y$,

$a^x{a}^y=a^{x+y},(a^x{)}^y=a^{xy},$

$(ab{)}^x=a^x{b}^x,{\left(\frac{a}{b}\right)}^x=\frac{a^x}{b^x},a^{-x}=\frac{1}{a^x}.$

The first identity is what the board checks. The others are the same bookkeeping: stacking powers multiplies exponents, and a product of bases splits across a common exponent. On this page $a>0$ (and $b>0$). Drop that and a negative base with a fractional exponent is not a real height.

The fixed point $(0,1)$ is $a^0=1$. Slide the base through $1$ and the curve flattens to the horizontal line $y=1$. The product law is still true there, and it says nothing new: every height is $1$. For $0<a<1$ the same law holds and the curve falls as $x$ grows. A positive integer exponent is the counting picture of that law,

$a^1=a,a^n=a\cdot a\cdots a$

with $n$ factors, and $a^2=a\cdot a$, $a^3=a\cdot a\cdot a$, and so on.

The opposite exponent is a reciprocal

The last identity in the list is not a new curve. Change the sign of the exponent you already dragged.

So $a^{-1}=1/a$, $a^{-2}=1/a^2$, and in general $a^{-n}=1/a^n$ for a positive integer $n$. The extra assumption is only the sign. Put the exponent back and you are on the first board again.

Only the nonnegative crossing

A positive integer power asks for a height. A root asks the other way: given a height $a>0$ and a positive integer $n$, which $x$ satisfies $x^n=a$?

The page keeps the answer in the half-line $x\ge 0$. There the equation has one solution, and that solution is written $a^{1/n}$. Older texts draw it as a radical, $\sqrt{a}$ or $\sqrt[3]{a}$. The power $a^{1/n}$ is the same number, and it still obeys the laws above, so the radical does not need a second rulebook.

The board freezes $n=2$. The curve is $y=x^2$ drawn only for $x\ge 0$, and the horizontal line is the height $a$. Their crossing is $\sqrt{a}$.

Drop $x\ge 0$ and an even power has another real solution, the negative one. This page still names only the nonnegative root. For a cube the negative axis is a different real solution of $x^3=a$ when $a<0$; the formula $a^{1/3}$ on this page is still the nonnegative one, and it assumes $a>0$.

Nested roots are the power law with fractional exponents. From

${\left(a^{1/n}\right)}^{1/m}=a^{1/(mn)}$

the radical form is $\sqrt[m]{\sqrt[n]{a}}=\sqrt[mn]{a}$. That is example 1: one identity, two notations.

Rational points crowd a real power

If the exponent is a ratio of positive integers, $x=m/n$, the root and an integer power are enough:

$a^x={\left(\sqrt[n]{a}\right)}^m.$

Together with $a^{-x}=1/a^x$, every rational exponent reduces to roots. A real exponent is the limit of that picture. Choose a sequence of real numbers $x_k$ with ${\lim }_{k\to \infty }{x}_k=x$. Then

${\lim }_{k\to \infty }{a}^{x_k}=a^x.$

That sentence is the continuity of this curve. Choose the $x_k$ rational and each $a^{x_k}$ is a power of a root; as $k$ grows, those heights approach $a^x$.

$\pi =3.141592\dots$, so the truncation $3.14=314/100$ is a rational stand-in:

$a^{3.14}=a^{314/100}={\left(\sqrt[100]{a}\right)}^{314}.$

More digits of $\pi$ give a closer height. The board places $3$, $3.1$, $3.14$, and $\pi$ on the same curve.

Read the curve backwards

Fix a base $a>0$ with $a\ne 1$. For every height $y>0$ the equation $y=a^x$ has one real solution $x$. That solution is the logarithm of $y$ to base $a$,

$x={\log }_{a}y.$

The first boards already drew $y$ as a height on $y=a^x$. The logarithm is the exponent you have to feed in to get that height. This board freezes $a>1$, so the curve rises and the flip across $y=x$ is a function. The point $(s,a^s)$ and the point $(a^s,s)$ are the same fact read in the two orders.

If $a=1$, every positive power is $1$, so a height other than $1$ has no exponent and the height $1$ has every exponent. There is no logarithm at base $1$. The condition $y>0$ is the other half: the curve never crosses the $x$-axis, so a negative height is not a power of a positive base.

The logarithm laws are the power laws with the coordinates swapped. For positive $c$ and $d$, and any real $x$,

${\log }_a(cd)={\log }_{a}c+{\log }_{a}d,{\log }_a\left(\frac{c}{d}\right)={\log }_{a}c-{\log }_{a}d,$

${\log }_a{c}^x=x{\log }_{a}c,{\log }_{a}a=1,{\log }_{a}1=0.$

The sum rule is the product rule from the first board: multiplying two heights adds the exponents that produced them. The integer points on that curve are the old table of a geometric row against an arithmetic row,

$\begin{array}{cccccc}1& a& a^2& a^3& a^4& \cdots \\ 0& 1& 2& 3& 4& \cdots \end{array}$

In 1544 Michael Stifel set those rows side by side in Arithmetica integra and saw that a product in the first row becomes a sum in the second. He also said he would rather notice the pattern than write the treatise. Logarithm tables are that correspondence, computed. The word itself is Greek, and it means a ratio. John Napier published a first incomplete table in 1614, with base $1/e$. After talking with Henry Briggs he agreed to base $10$. In 1617 Briggs published a base-$10$ table carried to $14$ decimal places. Those tables were part of the arithmetic behind Kepler's Rudolphine Tables. With machines that can evaluate the curve directly, the printed tables are no longer the tool they were.

One base carries every other

The natural logarithm is the case of base $e$, written $\ln y$ for ${\log }_{e}y$. For an arbitrary positive base and every real $x$,

$a^x=e^{x\ln a}.$

Knowing natural logarithms then gives every other base:

${\log }_{a}y=\frac{\ln y}{\ln a}.$

For $a=10$, $\ln a=2.302585\dots$ and $1/\ln a=0.434294\dots$. The number $e=2.718283\dots$ is the base that makes the exponential the natural one: from $y=e^x$ you read $x=\ln y$. Growth and decay make that base feel inevitable; the name "natural logarithm" is that reading.

The object on this page is $a^x$ for $a>0$, together with ${\log }_a$ when $a\ne 1$ and the height is positive. Integer powers count factors. Rational powers are roots. A real power, including $a^{\pi }$, is the limit of those rational powers along the same curve. The logarithm is the exponent that curve requires.

The next question is why an arbitrary real exponent is this limit, rather than a new kind of multiplication.