Constructing logarithmic functions from tables

0:00- [Narrator] We're told the table represents the logarithmic function f. So we see some inputs and some outputs. Complete the equation for f of x is equal to a log-base b of x. Use a subscript wherever necessary. So that's actually, that second part is really if you're inputting it on Khan Academy, but let's just try to figure out what a log base b of x is.

0:23So pause this video, have a go at that before we do it together. Okay, now let's just look at what's going on in this table right over here. So our inputs, we're not just incrementing by a fixed amount every time. It looks like our inputs are going from 3 to 9, 9 to 27, so it looks like we are multiplying by three for every successive input on this table over here, we are multiplying by three.

0:45That just keeps going all the way to 243. And when we multiply by three, what happens to the output? It looks like every time we multiply by three, our output goes up by two, our output goes up by two, our output goes up by two. So this is a pretty good clue here of what our base is.

1:04If our base were, say, log, if it was three, then every time we multiply by three, we should go up by a fixed amount, right over here. But let's just think about this a little bit. If this were just a simple, so let's just have x here, and if this was just log base 3 of x, what would this look like?

1:27Well, if x were 3, log base 3 of 3 is one. 3 to what power is equal to 3? It's the first power. If this were a 9, log base 3 of 9 is 2. If this were 27, then this would be a 3. 3 to the third power is 27. 81. You get the idea. 3 to the fourth power is 81, but that's not what we see right over here.

1:48We don't see 1, 2, 3, 4. We're not just incrementing by one. We have essentially twice that in every one of these situations. So really, if we just were to multiply this by 2, we would get our f of x. So if we were to just put a 2 here, log 2 times log base 3 of x, well then we get what we need.

2:08When x is 3, log base 3 of 3 is 1 times 2 is 2. Essentially all of these things times 2. So this would be 4, this would be 6, this would be 8, so on and so forth. So just like that, we have figured out that f of x is equal to 2 log base 3 of x. Now, this is a good answer and a nice simple answer if you just wanted a nice whole number base here, but it's actually not the only answer here.

2:39Another way you could have thought about it is, well, what happens if I multiply my input by 3, my output, my exponent? Well, what you could view as your exponent or maybe something being multiplied by the exponent goes up by a fixed amount. You could say, hey, what base if I multiply by 3, I now have to maybe square it or raise it to the power of 2 more in order to get to the next number.

3:06So you could have also tried something like this. You could have said, well, x, let's see, 3, 9, 27, so forth and so on. You could have said, well, if my base were the square root of 3, well then every time I multiply my input by 3, my exponent is going to increase by 2.

3:30So if I did this, this would've also worked because log base square root of 3 of 3, well, you have to square the square root of 3 to get to 3. To get to 9, you would've to go to 4. To get to 27, square root of 3 to the sixth power is indeed equal to 27. And so this in theory could be correct, but this one right over here is much simpler and easy to manipulate.