Exponentials and Logarithms

Outside mathematics, "exponential" is used to mean fast. That is not what the word means, and the substitution costs you the only property worth having. Exponential A function of the form f(x) = bˣ with a fixed positive base b, and the growth pattern it describes: one where the rate of increase is proportional to the amount already present, so twice as much gives twice as much new per step. That proportionality is the definition, and speed is not part of it: a pile doubling from 1 is behind a pile adding 3 for the first three steps and passes it at the fourth. Every such curve passes through (0, 1), is positive everywhere, and never turns round. growth is growth where the rate is proportional to how much there is right now. Twice as much stuff, twice as much new stuff per step. Nothing in that sentence mentions speed.

Two piles make the difference visible. Both start at 1. The left pile adds a fixed amount every step. The right pile doubles every step.

Set the fixed amount to 3 and watch the left pile win. At step 1 it is at 4 against 2. At step 2, 7 against 4. At step 3, 10 against 8. The pile that adds is ahead the whole time, and a reader who has been told exponential growth is fast has never seen it lose. At step 4 the doubling pile reaches 16 against 13 and passes for good. By step 10 the adding pile is at 31 and the doubling pile is at 1024.

Read the gain beside each column rather than the heights. The left pile gains the same amount every step, whatever it is standing at. The right pile gains its own current height. That second readout is the definition, printed as a number that changes, and it is the whole of what "exponential" means.

Now try to make addition win permanently. Raise the fixed amount to 10 and the adding pile leads for five steps instead of three, with the doubling pile passing at step 6. Raising the constant further moves the crossing later and never removes it. There is no setting of that slider where addition wins, because a fixed gain stays fixed and a gain equal to the current height does not.

Keep stepping and the drawing runs out of height. That is not the figure failing. Any fixed scale tall enough to show the adding column at all loses the doubling column within a few steps, and no scale exists that holds both for long.

Take a sheet of paper a tenth of a millimetre thick and fold it in half 42 times. Guess the thickness before reading on. Each fold doubles it, so the answer is 2⁴² · 0.1 mm, which is 439,805 km. The mean distance from the Earth to the Moon is about 384,400 km. Nobody can fold a real sheet 42 times, and the arithmetic is still the arithmetic.

Bacteria dividing is the same shape. Every cell present is a cell that can split, so the number of new cells per hour is set by the population already there, and the population is what sets its own speed.

Growth and decay are usually taught as two topics. They are one curve and one slider. In y = bˣ the number b is the base The fixed number being raised to a power, the b in bˣ and the subscript in log_b. It must be positive: a negative base flips sign between consecutive whole number exponents and has no real value at fractional ones. Above 1 the exponential curve rises to the right, below 1 it falls, and at exactly 1 it is the flat line y = 1 and has no logarithm at all, because nothing but 1 is ever produced. Three bases are in common use: 10 for tables, slide rules and every scale from decibels to pH, 2 in computing, and e in analysis., and every exponential curve you will meet is this curve with some b in it.

Above 1 the curve rises to the right, and the larger the base the harder it climbs. Below 1 the curve flips and falls to the right, which is decay, and it is not a separate family. (½)ˣ is 2⁻ˣ, the growth curve with its input negated, so it is the same drawing reflected left to right.

At exactly b = 1 the curve is flat. 1 to any power is 1, so the rule returns 1 for every input and the graph is the horizontal line y = 1. That is the boundary between the two behaviours and it is worth stopping on, because it is the one setting where the curve is not exponential at all.

Two things stay put whatever you do to the slider. The curve passes through (0, 1) for every base, because b⁰ = 1 for every b. And for any base either side of 1 the curve never reaches zero and never turns round: above 1 it flattens towards the horizontal axis on the left without landing on it, and below 1 it does the same on the right. That line it approaches and never meets is the horizontal asymptote A line a curve approaches arbitrarily closely without ever reaching. In the trigonometric graphs every asymptote is vertical and sits where the definition divides by zero: tangent and secant at the zeros of cosine, cotangent and cosecant at the zeros of sine. The general rule is that a reciprocal function has an asymptote exactly where the original function has a zero. An asymptote need not be vertical: 1/x has a vertical one at x = 0, where it has no value, and a horizontal one at y = 0, which it flattens towards as x runs off in either direction and never lands on. from the previous page, at y = 0.

The slider stops above zero and does not reach it, which is a fact about the domain The set of inputs a function accepts. It is part of the function rather than a note attached to it, so two rules with the same formula and different domains are different functions: 1/x excludes the single point 0 and √x excludes every input below 0, which are two different shapes of exclusion. Sine and cosine accept every real number. Tangent and secant exclude the angles where cosine is zero, and cotangent and cosecant exclude the angles where sine is zero, because a division by zero has no value there. rather than a limit of the figure. A negative base has no usable curve. (−2)³ is −8 and (−2)² is 4, so consecutive whole number inputs would flip the sign, and (−2)⁰·⁵ is the square root of a negative and has no real value at all. There is no curve to draw, so exponentials are defined for positive bases only.

Two to the what gives 1000? The question is ordinary and the notation for its answer is where people stall. 2⁹ is 512 and 2¹⁰ is 1024, so the answer is a little under 10. Written out, it is 9.9658 to four decimal places. That number is the logarithm The exponent that was missing. log_b y is the power b must be raised to in order to give y, so log₂ 1000 is about 9.9658 because 2 raised to that power is 1000. It is the inverse of the exponential with the same base, so its graph is that curve reflected in the line y = x, it accepts positive inputs only, and it passes through (1, 0). Its defining property, log(ab) = log a + log b, turns a multiplication into an addition and is what tables and slide rules were built on. of 1000 to base 2, written log₂ 1000, and it is nothing more mysterious than the exponent that was missing.

Every point on an exponential curve carries both questions at once. Go in from the horizontal axis, up to the curve, across to the vertical axis, and you have run the exponential: you gave an exponent and got a value. Go in from the vertical axis, across to the curve, down to the horizontal axis, and you have run the logarithm: you gave a value and got the exponent. The route reverses and the curve does not change. Reading a value off a curve in both directions is the procedure from the previous page, and the second direction now has a name.

One direction can fail and the other cannot. Every exponent has a value, so going in from the horizontal axis always arrives somewhere. An exponential curve above the axis never produces zero or a negative number, so going in from the vertical axis below zero arrives at nothing at all. Take the handle down there and watch the guide cross the whole picture without meeting the curve.

So the logarithm is the exponential undone. Two functions that undo each other have graphs that are mirror images in the line y = x, because undoing swaps which coordinate is the input and which is the output, and swapping a point's two coordinates is what reflecting in that line does. Run the fold and keep the exponential faintly in place, and the identity is a picture rather than a claim.

The reflection makes every feature of the log curve a consequence rather than a fact to memorise. The stretch of axis that carried no value is now on the horizontal axis, so the logarithm's domain The set of inputs a function accepts. It is part of the function rather than a note attached to it, so two rules with the same formula and different domains are different functions: 1/x excludes the single point 0 and √x excludes every input below 0, which are two different shapes of exclusion. Sine and cosine accept every real number. Tangent and secant exclude the angles where cosine is zero, and cotangent and cosecant exclude the angles where sine is zero, because a division by zero has no value there. is the positive numbers only. The exponential passes through (0, 1), so the logarithm passes through (1, 0). The exponential's flattening towards the horizontal axis becomes a vertical asymptote A line a curve approaches arbitrarily closely without ever reaching. In the trigonometric graphs every asymptote is vertical and sits where the definition divides by zero: tangent and secant at the zeros of cosine, cotangent and cosecant at the zeros of sine. The general rule is that a reciprocal function has an asymptote exactly where the original function has a zero. An asymptote need not be vertical: 1/x has a vertical one at x = 0, where it has no value, and a horizontal one at y = 0, which it flattens towards as x runs off in either direction and never lands on. at x = 0, where the log curve runs away without bound. And where the exponential climbs ever faster, the logarithm climbs ever more slowly, because the two axes have changed places.

One base has no logarithm and it is the one the last step paused on. At b = 1 the exponential returns 1 for every input, so no value except 1 is ever produced and the input that produced it cannot be recovered. Reflecting that flat line in y = x gives a vertical line, which fails the vertical line test A check on whether a drawing in the plane is the graph of a function of x. Place a vertical line anywhere: it selects one input and every output that input has, so if it meets the drawing more than once, that input has more than one output and the drawing is not a graph of a function. It is the definition read off the page rather than an extra rule. There is no matching horizontal test, because many inputs sharing one output is allowed.. There is no log base 1.

You have two six figure numbers and a pen. Multiplying them by hand is most of a page of working and around thirty separate chances to make a slip, and if you are an astronomer reducing a night's observations you have a hundred more waiting. Adding two six figure numbers takes one line and almost never goes wrong. So the useful question is whether a multiplication can be turned into an addition.

It can, and one identity An equation that holds for every value of its variable, as opposed to an equation solved for particular values. sin²θ + cos²θ = 1 is an identity; sin θ = 0.5 is an equation. The Pythagorean, reciprocal, quotient and cofunction identities are the four families that connect the six functions. does it. log(ab) = log a + log b. The logarithm of a product is the sum of the logarithms. It follows from what an exponent is: bᵐ · bⁿ = bᵐ⁺ⁿ, because multiplying m copies by n copies gives m + n copies. Read that backwards with the exponents as the unknowns and it is the identity above.

The procedure that falls out of it is mechanical. Look up log a. Look up log b. Add them, which is the easy step. Then look up which number has that logarithm, and that number is the product. Three lookups and one addition replace the page of working, and the arithmetic never gets harder as the numbers get bigger.

A pair of scales does the same thing without any lookups, because the lengths are the logarithms. Mark a scale so that the distance from its left end to the number x is log₁₀ x times the length of one run from 1 to 10. That puts 2 a little under a third of the way along, since log₁₀ 2 = 0.30103, and it squeezes 9 and 10 into the last twentieth, since log₁₀ 9 = 0.95424. Equal distances now stand for equal ratios instead of equal differences.

To multiply, slide the top scale until its 1 sits over a on the bottom scale. The top scale's origin has moved a distance of log a. Find b on the top scale, which is a further log b along. The point underneath it on the bottom scale is therefore at distance log a + log b from the bottom scale's left end, and by the identity that distance is log(ab). Read the number there and it is the product. Set 2 and 3 and read 6.

The number in the readout is worked out from where the scale actually sits, so a sloppy slide gives a slightly wrong product. That is honest and it is exactly what using one of these was like.

The history is two people and three dates. John Napier published logarithms in Mirifici Logarithmorum Canonis Descriptio at Edinburgh in 1614: 57 pages of explanation and 90 pages of tables, and the tables are logarithms of sines for angles from 0 to 90 degrees, aimed at the spherical trigonometry that navigation and astronomy run on. They are not base 10. Henry Briggs recast them, publishing logarithms of 1 to 1000 to fourteen places in Logarithmorum Chilias Prima in 1617, and 30,000 in all in Arithmetica Logarithmica in 1624, and it is his base 10 version that every table and every slide rule since has used.

The instrument came later and from two more people. Edmund Gunter, professor of astronomy at Gresham College, put a logarithmic scale along a two foot rule in 1620, and added lengths on it with a pair of dividers. Around 1622 William Oughtred set two of Gunter's scales sliding against one another, which removed the dividers, and that is the slide rule. It stayed the working calculating instrument of science and engineering for roughly three and a half centuries. Hewlett-Packard's HP-35, the first handheld calculator with logarithms and trigonometric functions on its keys, went on sale in February 1972. Keuffel and Esser made their last slide rule in 1976 and gave it to the Smithsonian.

A logarithm answers two questions at once: roughly how many digits a number has, and how many times bigger one number is than another. Both are questions about ratios rather than differences, which is what makes a logarithm the natural thing to put on an axis. On a linear axis equal distances are equal differences. On a logarithmic axis equal distances are equal ratios, so the same gap that carries 1 to 10 carries 10 to 100 and 100 to 1000, and each of those gaps is one order of magnitude A factor of ten, used as a unit of rough size: two quantities are an order of magnitude apart when one is about ten times the other. It is the integer part of a base 10 logarithm, which is why a logarithmic axis places each order of magnitude the same distance apart and can hold data spanning many of them on one page. "Within an order of magnitude" is a claim of agreement to within a factor of ten, not a claim of precision..

The straight line is the diagnostic worth learning. If y = A · bˣ then taking logs of both sides gives log y = log A + x · log b, which is a straight line in x with slope log b. So exponential data plotted with a logarithmic vertical axis comes out straight, and the steeper the line the larger the base. Anything else does not straighten.

The figure carries three datasets and they exist to separate two claims. One of them is the folded sheet from step 1, doubling once per fold, and it is the one that comes out straight. One spans a wide range without multiplying by any fixed amount, so the log axis opens it out and it stays bent. One spans under a decade and includes two dry days, which a log axis cannot place at all: zero has no logarithm, so those two points are not drawn.

Being readable on a log axis and being exponential are different claims, and that is the mistake to avoid. A log axis opens out any data that spans orders of magnitude, whatever shape it has. Straightness is the extra thing, and it is the only part that tells you the growth is exponential.

Several scales in the world are logarithms already, and each is the logarithm of a particular quantity in a particular base. Two of them are worth having in full, because they carry the two structural twists you will meet again.

The decibel is base 10 and it carries a factor in front of the logarithm. A power ratio in decibels is 10 · log₁₀ of the ratio, so 10 dB is ten times the power. Sound pressure level is 20 · log₁₀ of the pressure ratio, because intensity goes with the square of pressure and squaring inside a logarithm is doubling outside it. Its reference is 20 micropascals, which is near the threshold of human hearing.

pH is base 10 with a minus sign in front: pH = −log₁₀ of the hydrogen ion activity, or of its concentration in the version most people are taught. The minus is why the scale runs the way it does. pH 4 has ten times the hydrogen ion concentration of pH 5, so lower is more acidic.

Go back to the base slider. At every setting the curve is steeper where it is taller, because that is what exponential growth is. So there is a natural question: is there a base where the steepness is not merely proportional to the height but equal to it?

There is exactly one, and hunting for it is worth doing before being told. At base 2 the steepness comes out at about 0.693 of the height, so the curve is too lazy. At base 3 it is about 1.099 of the height, so the curve is too eager. Between them the two readouts converge, and at about 2.718 they agree.

Then move the point along the curve without touching the base. The ratio does not move. That is the part that makes the number worth a name: the match is not a coincidence at one lucky spot, it holds at every point on that one curve. On any other base above 1 the ratio of steepness to height is also the same everywhere, and it is only equal to 1 at this one.

That base is called e, and it is 2.718281828459045 to fifteen places. It is irrational, so the decimals never repeat and never stop. The logarithm to base e has its own name, the natural logarithm The logarithm to base e, written ln. e is about 2.718281828459045 and is irrational. It is singled out because y = eˣ is the one exponential curve whose steepness at every point equals its height there; for other bases the ratio of steepness to height is a constant other than 1, about 0.693 for base 2 and about 1.099 for base 3. Stating that requires a definition of steepness at a single point, which is a limit, so the fact is shown before it can be proved., and its own notation, ln, and it is the logarithm that appears in almost every formula in analysis for exactly the reason this figure shows.

Now the debt, stated openly rather than hidden. The figure prints a steepness, and this page has never defined what the steepness of a curve at a single point is. Steepness between two points is plain arithmetic: rise over run. At one point there is no run. What the figure does instead is take a short piece of the curve either side of the point and treat it as straight, which is an approximation, and the number it prints depends on how short a piece it takes. Widen that piece to the coarsest setting and the two readouts agree at 2.691 rather than at e.

So take e on the terms this page can support. It is the base where a measured steepness matches a measured height, at every point on the curve. That is a fact you can watch happen, and it is one honest definition away from being a theorem.

Put money in an account paying 5% a year, with each year's interest added to the balance so that the next year's interest is worked out on the larger amount. That is compound interest Interest worked out on a balance that already includes the interest added before it, so each period multiplies the balance by the same factor 1 + r rather than adding the same amount. After n periods the balance is (1 + r)ⁿ times the starting amount, which makes it exponential by definition: the amount gained each period is proportional to the amount already there. Interest that is paid out instead of added back is not compound, and grows by a fixed amount each period., and the question people actually ask about it is how long until there is twice as much. You could compute the balance year by year and watch for the moment it passes double. The question worth asking first is whether a logarithm can name that year before any of the years are counted.

After n years the balance is (1 + r)ⁿ times what you started with, because every year multiplies by the same factor. Twice as much means (1 + r)ᵗ = 2. The unknown is an exponent, which is the shape a logarithm exists for. Take the natural logarithm of both sides: t · ln(1 + r) = ln 2, so t = ln 2 / ln(1 + r). That is the doubling time How long something growing by a fixed factor each period takes to reach twice its starting size. Setting (1 + r)ᵗ = 2 and taking logarithms gives t = ln 2 / ln(1 + r), so the answer is a logarithm and does not depend on the starting amount at all. Reaching four times the start takes exactly twice as long, because log 4 = log 2 + log 2. The rule of 70, dividing 70 by the rate as a percent, is this formula with ln 2 rounded from 0.693 to 0.70 and ln(1 + r) replaced by r., and at 5% it is 0.693147 divided by 0.048790, which is 14.207 years.

The figure computes that number in the panel and then draws the years without it. Each dot is the dot before it multiplied by 1 + r, one multiplication per year, and that loop never reads the doubling time and never raises anything to a power. The ringed dot is the first one above the ×2 line, found by comparing values. At 5% the balance is 1.97993 times the start at the end of year 14 and 2.07893 at the end of year 15, so the ring sits on year 15. The panel named that year before any dot was placed, which is the check: the prediction and the count are two separate calculations and they agree.

The second dashed line pays off step 4. Reaching four times the start is doubling twice, and log 4 = log 2 + log 2, so the time to four times the start is exactly twice the time to double, at every rate. At 5% that is 28.413 years. It is log(ab) = log a + log b again, arriving as a statement about a clock.

The rule of 70 is the shortcut people actually use: divide 70 by the rate written as a percent. It comes from the same formula. ln 2 is 0.693147, and for a small r the value of ln(1 + r) is close to r itself, so the doubling time is close to 0.693147 / r. With the rate as a percent that is 69.3 divided by it, and 70 is used instead because it divides evenly by 2, 5, 7 and 10.

The approximation is excellent where the rate is small and drifts as the rate grows. At 2% the rule gives 35.000 years against a true 35.003. At 5% it gives 14.000 against 14.207. At 12% it gives 5.833 against 6.116, which is 0.283 of a year out. Turn the tick on and drag the rate, and you can watch a rule everybody quotes come apart at the top of its range.

This is one job and it is a narrow one. The formula answers "how long" for anything that multiplies by the same factor every step, and it answers nothing else. A rate that changes needs the calculation redone at every change, and the account here holds one rate for forty years because that is the only case this arithmetic covers.

One thing has gone undefined the whole way through. This page used the natural logarithm to predict a year, and the base it is built on was chosen in the last step by measuring a steepness the page never defined. Rise over run needs two points, and the steepness at a single point has none. What happens to rise over run as the two points are brought together, when the run is heading for zero and the ratio is heading for 0/0, is a limit. That is the next subject in this series, and e is the first thing it repays.

Every number on this page is worked out from the controls as you set them, including the ones in the legends, the arithmetic panel on the slide rule and the panel in the doubling figure. Nothing here was stored as a picture. The base is shared between the step about what the base does and the step that reads the curve backwards, so the same curve carries across; the hunt for e starts fresh at 2, and every other control belongs to its own figure.