Topic 2.10: Inverses of Exponential Functions

Thirty-three wins

This is a beach on Sagami Bay in Japan, with Mount Fuji across the water, and the thousand and twenty-four people standing on it are about to play the biggest game of rock-paper-scissors you’ve ever seen. The game is called the rock-paper-scissors train. You play one person, and if you lose, you grab the winner’s shoulders and join the back of their line, and from then on, whoever’s at the front plays for everybody behind them. Here’s the first round. Watch what the losers do.

That leaves five hundred twelve lines of two. Let’s play it in rounds, like a tournament, so in round two the people at the front play again, and every losing line joins the back of a winning one.

Round three makes lines of eight, and you can probably see where this is going, because every single round, every line on this beach doubles. As of 2026, there are about eight point two billion people alive. Now imagine every one of us played this game. How many games would the champion of the whole planet have to win?

By the end of this lesson, you’ll be able to work out how many wins it takes to be the champion of anything, whether it’s your school, a stadium, a city or the whole planet. And you’ll do it in your head. You’ll also see why doubling the size of the crowd only ever costs the champion one more win, so let’s start with the thousand and twenty-four people on this beach, and play their game all the way to the end.

Ten wins

Round four makes lines of sixteen, then thirty-two, sixty-four, a hundred twenty-eight, two hundred fifty-six, five hundred twelve, and in round ten, the last two lines play each other. And there’s our champion, at the front of one single train of all one thousand twenty-four people, and it took exactly ten wins.

After zero rounds the champion was a line of one, then two, four, eight, sixteen, and every round doubled it again, all the way to a thousand twenty-four. So if we call the number of rounds x, the champion’s line after x rounds is two to the x people long, and it starts at one, because before anybody plays, everybody is a line of one.

Now let’s double the crowd. Here comes a second group of a thousand and twenty-four, who’ve already played their own game to the end, so now there are two champions, and two thousand forty-eight players on the beach.

How many rounds does it take, from the very start, to crown one champion of all two thousand forty-eight? So doubling the crowd only added one single game. Times two in the players is plus one in the wins, and you’re going to see that rule again and again today. These numbers came from the beach. Let’s take them over to the board.

Asking it backwards

Here’s the champion’s line from the beach. The rounds are on the left, and the length of the line is on the right, and every time the rounds go up by one, the line doubles. This table answers the forward question, after so many rounds, how long is my line? But the question everybody on that beach really cares about runs the other way, there are this many of us, so how many games do I have to win?

So let’s swap the two columns. Watch where every number ends up. The line lengths moved over to become the inputs, the rounds moved over to become the outputs, and the plus ones and times twos came along for the ride. This backward function is called the logarithm base two, or log base two for short, so log base two of thirty-two is five, because a line of one takes five rounds of doubling to reach thirty-two.

The two is how fast a line grows. The thirty-two is how long it ends up, and the five is how many rounds it took. Every pair from the old table shows up flipped, so five comma thirty-two became thirty-two comma five, and in general, if s comma t is on two to the x, then t comma s is on log base two of x.

That’s what makes two to the x and log base two of x inverse functions. Each one answers the other’s question backwards. And look at the marks now. On the exponential, adding one to the input multiplies the output by two, and on the logarithm, multiplying the input by two adds one to the output.

That’s the rule from the beach, double the crowd and the champion needs one more win. On the AP exam you’ll see it stated like this. A logarithm’s outputs change additively while its inputs change multiplicatively.

Thirty-three

So now we can answer the question from the beach, in our heads. Two to the tenth is one thousand twenty-four, just about a thousand, so every ten wins multiplies the crowd by about a thousand. Ten wins makes you champion of about a thousand people, twenty wins about a million, and thirty wins about a billion.

So a class of thirty needs five wins, because thirty is more than sixteen but less than thirty-two, and a school of two thousand needs eleven, because it’s just under the two thousand forty-eight on the beach. One more trick before you try one. Two to the seventh is a hundred twenty-eight, so a crowd about a hundred times bigger takes about seven more wins.

Here’s one for you, in your head, no calculator. Michigan Stadium holds one hundred seven thousand six hundred one people, so how many wins would it take to be champion of the whole stadium?

And now the whole planet. Eight point two billion is about eight times a billion, a billion takes thirty wins, and eight is three more doublings, so that’s thirty-three. A calculator says log base two of eight point two billion is about thirty-two point nine, and since nobody can win point nine of a game, the champion of Earth needs thirty-three wins.

If you guessed thirty-three at the very start, you were already thinking in logarithms. Two to the thirty-third is about eight point six billion, more people than there are, so about three hundred ninety million lucky people would get to sit the first round out.

Through the line

The sun is going down now, and the first six trains from the game, of one, two, four, eight, sixteen and thirty-two people, are standing on a graph drawn in the sand. From straight above, each train stands at its number of rounds. Look at the person at the front of each one. They’re standing right on a point of two to the x.

To the left, where there aren’t any whole people, the curve keeps going, through a half and a quarter, getting closer and closer to the x-axis in the sand without ever touching it. Now let’s light a row of lanterns along the line y equals x, where both coordinates are equal, and a few people, like the one at two comma two, are already standing right on it.

Everyone is about to walk straight across the lanterns and land just as far on the other side. Plant the flag where you think the front of the train of eight, at three comma eight, will land.

Now everybody crosses the line, all at the same time. Every column is lying down as a row now. And the people at the front landed exactly on the curve of log base two of x.

Let’s run it back. Everyone walks back to their column. Watch it one more time. And this time, keep your eyes on the table in the corner. The table swapped its rows at the very same moment the people crossed, so swapping the pairs in a table and reflecting a graph over y equals x are the same move.

Before, each step to the right doubled a column, and now each step up doubles a row. Even the line the curve was hugging, y equals zero, swings through the lanterns and stands up as x equals zero, so the floor has become a wall. Two to the x takes any input and gives only positive outputs, so log base two of x takes only positive inputs, and gives any output. Let’s lift this picture off the sand and onto paper.

The mirror, on paper

Here’s two to the x with the line y equals x, and on paper we can use any input at all, even negative one, which gives a half. Swap every pair and the whole curve crosses the line, so negative one comma a half lands on a half comma negative one. Zero comma one lands on one comma zero, and every log like this goes through one comma zero, because every line starts as a line of one.

Here too the floor becomes a wall, as y equals zero turns into x equals zero, so the log only takes positive inputs. A lot of people hear the word inverse and think log base two of x means one over two to the x. But that’s a half to the x, the mirror image over the y-axis, which is the wrong mirror. An inverse swaps the inputs and the outputs, so it always means the diagonal mirror.

There and back

Inverse functions undo each other, and you can feel it in the game. Win five games and your line is thirty-two people long, then ask how many rounds it takes to make a line of thirty-two, and you’re right back at five. It works in the other order too, log base two of thirty-two is five, and two to the fifth is thirty-two again.

So in general, log base b of b to the x is x, for every x, and b to the log base b of x is x, for every positive x. Why only positive? Try log base two of negative four, and you’re asking how many rounds it takes to make a line of negative four people, and there’s no such thing, because two to any power at all is positive. Here’s one to try without a calculator. What is three to the power log base three of seven?

Players per game

So far everybody has played games of two. Now suppose every game has three players, and both losers join the winner’s line, so every line triples each round. Both curves bend together in the mirror, and the champion of Earth would need only twenty-one wins. With ten players a game it’s ten wins, and in a hundred-player battle royale, the champion of the whole planet wins just five matches.

And with one player a game, nobody ever loses, so the lines never grow, and the exponential lies flat at a height of one. Its mirror image stands straight up as the line x equals one, which isn’t even a function, and that’s exactly why the base of a log can never be one.

With a base of a half, the exponential gives the fraction of players still unbeaten, a half after one round and a quarter after two, so it falls, and its mirror image falls too. Now you take the players-per-game slider at the bottom, and find the smallest game that crowns the champion of Earth in just five matches.

All of these are logarithmic functions, and the general form is f of x equals a times log base b of x, where b is positive and not one, and a is not zero. A base of one is the game that never ends, and with an a of zero, every output would just be zero. The base is the players per game, and a stretches every output, like the seconds each game takes.

At five seconds a game, crowning a champion takes five times log base two of the crowd, so a thousand twenty-four players take fifty seconds, and doubling the crowd adds five seconds, not fifty. The whole planet would take under three minutes, if nobody ever tied. On the graph, a equals two doubles every output, but one comma zero stays put, because two times zero is still zero. And a negative a flips the graph over the x-axis, so now the curve falls instead of climbing.

On the exam

Here’s exactly how the exam likes to ask this. You get a table of some function h, and you have to decide whether h is logarithmic, and if it is, find its formula. Each input is three times the one before. And each output is two more than the one before. When the inputs multiply while the outputs add, that’s the signature of a logarithm.

h of one is zero, just like every log in this form, and the inputs multiply by three, so let’s use base three. At x equals three, log base three of three is one, so h of three is just a, which the table says is two. So, h of x is two times log base three of x. So what is h of eighty-one?

And here’s the other classic. The graph of g of x equals b to the x goes through the point three comma sixty-four, and f is the inverse of g, so find f of x, then f of sixteen and f of a half. The point says b cubed is sixty-four, and four times four times four is sixty-four, so b is four.

So g is four to the x, and its inverse is f of x equals log base four of x, and the point three comma sixty-four on g is sixty-four comma three on f. f of sixteen is two, because four squared is sixteen, and f of a half is negative a half, because four to the negative one half is one over the square root of four, which is a half.

The same thirty-three

So here’s the whole lesson. To invert an exponential, swap every pair, which reflects its graph over y equals x and turns the floor into a wall. Times two in the input of log base two is plus one in its output, and the general form is a times log base b of x.

It’s dark on the beach now. Everyone in the one long train is holding a light, with the champion of all two thousand forty-eight at the front. Pick one secret person in that train, and ask yes-or-no questions about them. Is your person in the front half? Every answer puts out half the lights.

Two thousand forty-eight, then a thousand twenty-four, five hundred twelve, and on down, until after eleven questions one light is left. It’s the game run backwards, eleven questions for eleven wins, so for the whole planet it’s the same thirty-three again. Thirty-three yes-or-no questions can find anyone on Earth.

Harvard’s David Malan shows this by tearing a phone book in half, and as he puts it, if you double the phone book next year from four billion to eight billion, thirty-three steps instead of just thirty-two. Whenever a crowd doubles, the answer goes up by just one. And now you can work out how many wins it takes to be the champion of anything, in your head.