Exponential and logarithmic functions, and the two derivatives that make them worth having

0:00This topic begins with a warning about itself, and the warning is worth taking seriously. Much of what follows, it says, is motivation rather than proof. Two of the results here are argued from first principles. The rest are asserted and left standing. That is not a complaint. Motivation is a perfectly good thing for a first pass to offer. But it changes what you should be ready to reproduce, and what you should simply learn.

0:32So this video keeps a running note of which is which. Everything you are about to see falls into one of two boxes: argued, or asserted. And where something is only asserted, we will check it by computation instead, because asserted is not the same as false. By the end you will have two lists, and the shorter one is the one you can be asked to prove. Start with growth.

1:00Put the input, its square, its cube and its fourth power on the same pair of axes. The usual claim is simple: the higher the exponent, the steeper the climb. To the right of one, that is exactly right. At an input of two, the fourth power has reached sixteen while the input itself has only reached two. But look to the left of one, because that region gets drawn too.

1:28At an input of a half, the fourth power is one sixteenth, and the input is a half. The order has completely reversed. The fourth power is now the flattest of the four, not the steepest. This video checked five sample inputs between nought and one: at every one of them the four powers run in the opposite order. And five inputs beyond one: at every one of those they run in the order you were promised.

1:58All four curves pass through the point at one and one, and that is where the order turns over. That marked dot is not decoration. It is the hinge. Here is the comparison that makes the point without evaluating anything. Take the tenth power and the fifteenth, and let the input run from one up to two. The tenth power arrives at two to the tenth. The fifteenth arrives at two to the fifteenth.

2:28You do not need either number. The second is the first multiplied by two, five more times, which is thirty-two times as large. That is exact, and this video checked it: two to the fifteenth divided by two to the tenth is thirty-two, on the nose. Five more factors of two, and a climb thirty-two times higher over the same run of inputs. That is what a higher exponent buys you, to the right of one.

3:01Now bring in a different kind of function. Not the input raised to a fixed power, but a fixed base raised to the variable power. Ten to the x, say. And the claim made for it is a big one: this eventually outruns every power, no matter how large the exponent. The evidence offered for that claim is one calculation, at one input. Take the input to be a thousand.

3:29The hundredth power of a thousand is ten to the three hundred. Ten raised to a thousand is ten to the thousand. Both of those are exact, and both were checked here. Neither can be written out. Ten to the three hundred has three hundred and one digits, and that is itself the lesson: leave them as exponents and compare the exponents. Three hundred against a thousand. Ten to the x wins, and wins enormously.

4:01And then comes a second warning: the general claim will not be proved. Fair enough. But watch carefully what that one check does and does not establish. Here is the trouble with one case. Take the hundredth power and compare it against ten to the x at every whole input from two up to two thousand. This video did exactly that, in exact whole-number arithmetic, with no logarithms involved anywhere.

4:33At two hundred and thirty-six of those inputs, the hundredth power is the larger one. And not by a whisker. At an input of two hundred, the hundredth power is around ten to the two hundred and thirty, against ten to the two hundred. So there is a long stretch where the exponential is losing, and the single check at a thousand sits far beyond the end of it. To see how little that check settles, take three different readings of the claim.

5:05First reading: the exponential leads at every whole input above one. Second reading: it leads at every whole input above a hundred. Third reading: it leads from two hundred and thirty-eight onward. At the input of a thousand, all three are right. The one check cannot tell them apart. Across the whole sweep of nineteen hundred and ninety-nine inputs, the first reading is wrong two hundred and thirty-six times, the second is wrong a hundred and thirty-seven times, and only the third is right every single time.

5:44One case agreed with the truth and with two falsehoods equally. That is what an illustration is. And that window is not an accident of the hundredth power. It is there for every exponent large enough, and it grows with the exponent. For the first four powers there is no window at all. Ten to the x leads from the very start. For the tenth power the window runs from two to nine, and then something neat happens.

6:16At an input of ten, ten to the tenth and the tenth power of ten are the same number. Exactly level, neither ahead. For the fifteenth power the window runs from two to nineteen. For the hundredth, from two all the way to two hundred and thirty-seven. Every one of those edges was found by exact comparison of whole numbers, not by estimating. So the pattern is this: the bigger the exponent, the longer the exponential spends behind, and the further out you have to look before the claim becomes visible at all.

6:54Which is precisely why a single input, however large, is the wrong instrument for this job. The general claim can actually be settled, and it takes three lines. Compare the exponents rather than the numbers. Ten to the x has exponent x. The hundredth power of x has exponent a hundred times the logarithm of x at base ten. So watch the gap between them: x, less a hundred times that logarithm.

7:27Its slope, which we will be able to measure properly by the end of this video, is the natural logarithm of ten, less a hundred over x. That slope is negative while the input is small, and it turns positive as soon as a hundred over the input drops below the logarithm of ten. Forty-four is the first whole input where that happens, and this video pinned it down: a hundred over forty-four is below the logarithm of ten, and a hundred over forty-three is above it.

8:03So from forty-four onward, the gap can only grow. And at two hundred and thirty-eight the gap is already positive. Ten to the two hundred and thirty-eight really is bigger than two hundred and thirty-eight to the hundredth, exactly. A quantity that is positive at one point and only increasing afterwards stays positive for ever. That is the entire proof, and it needs the derivative of a logarithm, which is the second half of this video.

8:36Which is why the two halves come in this order. So, the definition. A fixed base above one, raised to the variable power. That is an exponential function. It comes with five features, and every one of them was measured here at three different bases: two, ten, and one more we are about to meet. One. The domain is the whole line. Every real input is allowed, negative ones included.

9:07Two. The range is the positive numbers. The value is never nought and never negative. Three. The point at nought and one is always on the graph, because any base to the zeroth power is one. Four. The graph rises from left to right, at every base above one. Five, and this is the one worth measuring rather than repeating: for large negative inputs the curve draws close to the horizontal axis without ever meeting it.

9:41At an input of minus forty, the value is smaller than a millionth of a millionth, and it is still not nought. A curve can get arbitrarily close to a line and never touch it, and that line has a name we will use: an asymptote. One base matters more than all the others, and it does not arrive as a number. It arrives as a sum. Add one, plus one, plus a half, plus a sixth, plus a twenty-fourth, and onward: the reciprocals of the factorials, for ever.

10:17Those partial sums climb, and they climb towards something. This video summed the first thirty of them. Every one lies strictly between two and three, and each is larger than the one before it. The number they climb towards is the base of the natural exponential function. It is written e. The exponential built on that base is the one the rest of the mathematics will use. And the reason why will not be visible until the last few minutes of this video.

10:52Now turn the whole thing round. If a base carries an exponent and reaches a number, then given the base and the number, that exponent deserves a name. It is called the logarithm. That is the entire definition. A logarithm is an exponent, read backwards. Two cubed is eight, so the logarithm of eight at base two is three. Ten to the fourth is ten thousand, so the logarithm of ten thousand at base ten is four.

11:24And now the best thirty seconds in the whole topic. Six hundred and twenty-five is five to the fourth. So its logarithm at base five is four. But six hundred and twenty-five is also twenty-five squared. So its logarithm at base twenty-five is two. One number, two bases, two different answers. And the second is exactly half the first, because twenty-five is five squared. Change the base, and the exponent changes with it. There is a rule hiding in that, and we will prove it shortly.

12:02All four of those conversions were checked here, in both directions. Make the logarithm a function, and something very tidy happens. The exponential takes the whole line in, and gives the positive numbers out. The logarithm takes the positive numbers in, and gives the whole line out. The two sets have simply changed places. This video measured that exchange in both directions. Every input the exponential accepts, negative ones included, comes back as a value the logarithm takes.

12:41Every positive number the exponential can reach is an input the logarithm accepts, and nothing at nought or below is. The logarithm's own six features all follow from that swap. Its domain is the positive numbers, its range is the whole line, it passes through the point at one and nought, and it rises from left to right. And near nought it drops below every bound you care to name. We tested three depths at three bases: name any depth, and there is an input above nought where the logarithm is deeper than it.

13:21Then one convention, easy to miss and expensive to miss. From here on, a logarithm written with no base at all means the natural one. Not base ten. That single convention decides the derivative you will meet in a few minutes. Draw the two curves on one pair of axes and the exchange becomes something you can see. The exponential passes through the point at nought and one. The logarithm passes through the point at one and nought.

13:54Those are the same pair of numbers, in the other order. Draw the diagonal line between them, and each curve is the other's mirror image in it. That is what an inverse function looks like. Reflect the graph in the diagonal, and the arrow reverses. The domain of one is the range of the other, which is the same sentence again, read geometrically. They are not two different kinds of object. They are one object, seen from two sides.

14:28Now the first of the two things this topic actually proves, and it takes five lines with no calculus in any of them. Suppose the logarithm of a number at one base is p. Suppose at a second base it is q. And suppose the logarithm of the first base, taken at the second base, is r. Turn all three of those into power statements, and then substitute one into another.

14:56You get the second base carrying the product of p and r, and the same second base carrying q. Same base, same value, so the exponents must agree. So q is p times r, which rearranges into the rule: the logarithm at the first base is the logarithm at the second, divided by the logarithm of the first base at the second. Exponent arithmetic from start to finish. Not a limit in sight.

15:28And to be sure that is the right arrangement rather than a plausible one, this video scored it against four rivals: the same quotient upside down, the product, the difference and the sum. Thirty-six pairings, and the truth in every case was the exponent itself, hunted down by repeated halving, so a quotient of logarithms was never checked against another quotient of logarithms. The stated quotient answers all thirty-six. The one turned upside down answers five, all of them cases where the answer happens to be its own reciprocal. The product, the difference and the sum answer none at all.

16:16The second thing this topic proves is the logarithm of a product, and it uses exactly the same method. Write each number as the base carrying its own logarithm. Multiply the two together. Powers of a common base add their exponents. So the exponent of the product is the sum of the two exponents. Which says, in the other language: the logarithm of a product is the sum of the logarithms.

16:46Again, no calculus. Just indices. And again it was scored: thirty-six pairings of six numbers with themselves, judged against a logarithm computed from a series that knows nothing about any identity. The sum answers all thirty-six. The product of the two logarithms, their difference, and the first logarithm plus the second number answer none at all. The logarithm of the sum answers exactly one, and it is the single pair whose sum and product are the same number.

17:23That is the whole of the argued half of this topic. Two results, both from exponent arithmetic. Everything else here is asserted. Three specialisations follow immediately from the product rule, and are handed to you rather than argued. The logarithm of a square is twice the logarithm. The logarithm of a whole power is that power times the logarithm. The logarithm of a quotient is the difference of the two logarithms.

17:57And then a fourth, which is a genuinely stronger claim: that the power form holds for any real exponent, not only whole ones. Asserted, and explicitly not argued. All four are true. This video checked each of them at six numbers, with exponents of a half, of minus five thirds, and of twenty-two sevenths among them. So what is missing is an argument, not a fact, and it is worth knowing which of the two you are short of.

18:30There is also a small trap in how those identities are usually written down. Every logarithm in them carries its base as a subscript, except the last one on the right, which is often set with no base at all. But we have just agreed that no base means the natural one. Read exactly as written, that identity holds at the natural base and fails at every other base tried, and this video tried four.

19:00Write the base in. It costs one character. Here is a question that looks like it has an obvious answer. Take the logarithm of an input, then exponentiate the result. Do you get the input back? The reflex says yes, of course: the two operations undo each other. The reflex is half right, and the half that is wrong is the half that matters. Take the exponential first and the logarithm second, and you do get the input back, at every input, negative ones included.

19:37That works because the exponential accepts everything and hands the logarithm something positive to work on. Take the logarithm first, and at any input that is nought or below you have already stopped, because there is no logarithm there to take. This video ran both orders at eight inputs. Logarithm first: four of the eight come back, and the other four have no meaning at all. Exponential first: all eight.

20:09So the two orders are not the same operation. One of them is total and the other is not, and the difference between them is exactly the logarithm's domain. Which brings us to the two derivatives the rest of the subject runs on. The natural exponential is its own derivative. And the natural logarithm differentiates to the reciprocal of the input. Both are stated here, and neither is derived here. The argument for them lives elsewhere, and it is more honest to say so than to invent one.

20:48So they were measured instead, from the definition, as limits of difference quotients. The exponential first, at eight inputs, against four rivals: the power rule applied by reflex, the exponential over the input, the exponential one step back, and the input times the exponential. The exponential itself answers all eight. Two of those rivals answer nothing at all, and the other two answer exactly one input each: the input of one, where multiplying by it and dividing by it both change nothing.

21:27Then the logarithm, at seven inputs, and here the base convention does all the damage. The reciprocal answers every one of the seven. And the reading a student reaches for when a bare logarithm is taken to mean base ten, one over the input times the logarithm of ten, answers none of them. That is not a small error. It is the wrong function. The same rejected formula is the exactly right derivative of the logarithm at base ten, at all seven of those inputs.

22:03So the mistake is never in the differentiating. It is in which logarithm you thought you were looking at. With those two derivatives in hand, everything else is the chain rule. Fifteen worked items were checked here, every one against its own difference quotient rather than against the algebra that produced it, and all fifteen survive at fifty inputs between them. And that test is not accepting everything: the same answers a thousandth too large, or with the sign turned over, or a thousandth adrift, are refused at every one of those fifty inputs.

22:46Two of the fifteen are worth doing slowly. The first is a sum of five exponentials: of the input, of its square, of its cube, of its fourth power and of its fifth. Differentiate term by term and the factors that appear are one, two, three, four and five. That pattern is invisible until all five terms are written out, which is the only reason to write them all out.

23:14The second is the square root of the exponential of a square root. Attacked head on, that is a chain of four stages and about four lines. But a square root is a power of a half, and a power of a power multiplies the two exponents. So the whole thing is just the exponential of half a square root, and this video checked that rewrite as an identity rather than assuming it.

23:42Now it is a chain of two stages, and the answer is itself, over four times the square root. Simplify before you differentiate. Four lines become one. So, the two lists. Argued: the change of base, and the logarithm of a product. Both from exponent arithmetic, both reproducible on paper in five lines, and both entirely fair to be asked for. Asserted: the growth story, the graph features, the three specialisations of the product rule, the real-exponent version, and the two derivatives.

24:23All of them true, and none of them argued here. Knowing which list a result sits on is the difference between studying and guessing. Three things to carry away. The first: one case is not a proof. The hundredth power really does lead ten to the x at two hundred and thirty-six whole inputs, and a single check at a thousand agreed with the truth and with two falsehoods equally.

24:52The second: a bare logarithm means the natural one. Read it as base ten instead and every derivative from here on is wrong by a constant factor. The third, and the one that generates all the rest: a logarithm is an exponent. Every identity in this topic is a fact about exponents wearing different clothes, and if you can turn a logarithm statement back into a power statement, you can rebuild all of them for yourself.