The exponential and the logarithm, their domains, ranges and graphs

0:00There is a single number that both of the functions in this video are built on. It is written e. And what gets said about it, at this point, is that it lies somewhere between two and three. That is the whole of the claim. No decimal expansion, no formula, no construction of it at all. That is not a gap somebody forgot to fill in. It is the reason the next thing along has to be argued with an inequality rather than computed with a formula.

0:33So before anything else: what does between two and three actually pin down? A statement pins down a decimal place when both of its ends agree there. Between two and three fixes zero decimal places. Not the first one. Not any of them. This video's own arithmetic, which is a different thing from that claim, brackets the same number to twelve places: two point seven one eight two eight one eight two eight.

1:05Twelve against none. If you already knew those digits, you brought them in from somewhere else, and it is worth knowing that you did. One more reading, because the next topic leans on it. Take two away, and what is left is zero point seven one eight two eight one eight two eight. Small, and firmly above nothing. The first function is the rule that sends a number to e raised to the power of that number.

1:35Two things have to be said about any rule before it is a function: what you are allowed to put in, and what comes out. What you are allowed to put in is every real number there is. Here that is a measurement and not an assertion. Eleven places were put to the machinery that computes this, running from twenty-five below nothing to twenty-five above it, and the number it declined to answer for is zero.

2:04That zero is only worth something if the machinery is capable of declining. So ask it the same eleven questions with only three terms of its series to work with. Now it declines ten of the eleven. The domain is not being waved through. It is being asked. Now the other half. What comes out. Of those same eleven readings, the number landing below nothing is zero, and the number landing exactly at nothing is zero.

2:38Eleven out of eleven are above nothing. And they climb. Walking the eleven places in order, the number of steps that fail to be a rise is zero. Read the same run off the same curve with its sign turned over, and every one of the ten steps fails. So the values are the positive numbers, and the curve rises across the whole of the line it is defined on.

3:07Go the other way, to the left, and something happens that is worth being precise about. The drawing shows a curve running close above the flat axis and never touching it. Two separate things are being claimed there, and they usually get run together. First, that it gets arbitrarily close. Seven bounds were offered - a tenth, a hundredth, a thousandth, a ten-thousandth, a millionth, a hundred-millionth and a ten-billionth - and the number of them the readings never get below is zero.

3:42You can even say where each one is passed. A tenth and a hundredth are beaten by five to the left. A thousandth and a ten-thousandth by ten. A millionth by fifteen, a hundred-millionth by twenty, and a ten-billionth by twenty-five. Second, that it never arrives. Of those same readings, the number sitting at or below nothing is zero. Under every bound, and at nothing not once. Neither of those zeros is a formality, and here is how you know.

4:17Take the identical curve and move it down by three. Put the identical two questions to it. Now six of the readings are at or below nothing. That curve crosses. Same machinery, same bounds, different answer. The zeros are readings. There is one place on this curve worth marking, and it is where the power is nothing. Anything raised to the power nothing is one, so the reading there is exactly one, in a bracket of no width whatever.

4:51The drawing meets the upright axis at height one. Hold on to that. It is going to come back in a few minutes wearing a completely different face. Now the second function, and the thing that makes this pair worth spending time on. It is not given by a formula. It is given by an equivalence. The logarithm of x is y - exactly when e to the power y is x.

5:19Read that from left to right and it tells you what the logarithm of a number is. Read it from right to left and it tells you the same thing about a power of e. There is no third statement anywhere. That one line is the entire definition. Which means the honest way to get a logarithm is not to evaluate anything. It is to go looking for the power.

5:45That is exactly what the checking behind this video does. There is no logarithm function anywhere in it. There is a search. Given a number, go and find the power of e that lands on it, and report what you found - or report that you found nothing. Nine numbers were handed to that search. Two below nothing, nothing itself, and six above it. It found a power of e for six of them, and refused three: minus four, minus one, and nothing.

6:20Those six are the logarithm's domain, and it matters a great deal how they were arrived at. There is no line anywhere in that search asking whether a number is below nothing. Not one. The refusals are simply what happens when the looking fails. Here is how you can tell that is true. Hand the same search a different function, and watch what domain it reports. Hand it a cube, whose values are every number there is. It finds all nine and refuses none - the negatives included.

6:56Hand it this same exponential moved down by three. Now it reports minus one as inside the domain, and refuses only minus four. Hand it minus one over the exponential, whose values are everything below nothing. The answer comes out the other way round: two found, seven refused, and the two are the negatives. Hand it a function with only one value, and it finds nothing at all. Four different domains, one line of code.

7:29So the positive numbers are a fact about the exponential. They are not a rule about logarithms. And the refusals are not all the same refusal. The search can fail in two ways, and they are two different sentences about the function. It can fail because the values never come down that far. That is what happens at nothing for the exponential. Or it can fail because the values never climb that high. That is what happens at nothing for the turned-over one.

8:05Neither sentence mentions a sign anywhere in it. So the logarithm not being defined at nothing is a consequence of where the first function's values are. It is not an extra rule bolted on afterwards. Ask the search for the power of e that lands on one. It comes back with a bracket straddling nothing - too narrow, in fact, for a sign to be read off it at all.

8:34The answer is nothing. The logarithm of one is nothing. And that is the same statement as e to the power nothing being one. Not a similar statement. The same one, read the other way through the equivalence. One drawing crosses the upright axis at height one. The other crosses the flat axis at one. Two crossings, one fact. That doubling is not a coincidence, and it is the whole reason the domains and the value sets are swapped rather than merely different.

9:10The equivalence says a pair of coordinates sits on one drawing exactly when the pair with those two coordinates swapped sits on the other. Swapping the two coordinates of every point is reflection in the line y equals x. Measured: each of the eleven points of the first drawing was taken, its coordinates swapped, and the swap tested against the second drawing. The number that miss is zero. Test those same swaps against the drawing they came from, and all eleven miss.

9:46The same thing again, as a journey. Out through the exponential and home through the search: of eleven places, the number failing to come back where they started is zero. Ask them to come home at the negative of where they started, and ten of the eleven fail. And start the journey off the turned-over curve instead, whose values no power of e ever reaches, and all eleven fail to come home at all.

10:18So what does the logarithm actually take as its values? Over the six numbers a power was found for: two readings below nothing, one at nothing, and three above it. Both signs, and nothing itself. That is the exponential's domain, arrived at from the other end. And coming in toward nothing from above, the readings run away downwards with no floor under them. At a tenth, the search reads minus two point three zero two five eight five. At a hundred-millionth, minus eighteen point four two zero six eight.

10:58Six bounds going downwards were offered. Walking in that far gets past five of them and not the sixth. Go one more decimal place in, to a thousand-millionth, and the reading is minus twenty point seven two three two six five, which passes the sixth as well. There is no floor. There is only how far in you were willing to walk. That is worth separating from something it gets confused with.

11:28Walking in toward nothing, the logarithm's readings pass every bound you care to set. They run away. Standing on nothing, there is no reading at all. The search refuses. Those are not the same failure. It is very large and negative is a statement about a walk. It is not defined is a statement about a single place. And the second one is not a decree handed down. It is what the first function's values leave over.

12:02One last check, because this pair leans on a rule that came before it. e to the minus x, against one over e to the x: across all eleven places, the number that disagree is zero. Leave the minus off and ten of the eleven disagree. And the number of readings the reciprocal route refuses to divide by is zero - which is the claim about the values again, arriving from a third direction. Nothing here divides by nothing, because nothing here is nothing.

12:38So here is what is on the table. One function from the whole line to the positive numbers, rising, never arriving at the axis it approaches, crossing at height one. One function from the positive numbers to the whole line, defined by an equivalence rather than a formula, with no floor as it comes in and no value at all at nothing. Each one's inputs are the other's outputs, and the two drawings are mirror images in the line y equals x.

13:13And the number they are both built on has not been constructed anywhere. It has been bracketed between two and three, which fixes not a single decimal place. That is deliberate, and it is the point. What comes next cannot be got at by evaluating anything. It has to be squeezed - and the quantity it gets squeezed with is the one measured right at the start: what is left when two is taken away, small, and above nothing.