 In this video I want to talk about the change of base formula for logarithms and gives us a way from changing from one base to another So suppose we have two acceptable bases like the so a and b so a is greater than zero But not equal to one b is greater than zero but not equal to one Those are both acceptable bases Then it turns out that the log base a of x is equal to the log base b of x divided by the log base b of a So b is your new base. You can pick whatever you want The old operand goes on top the old base goes on the bottom and this gives us the so-called law The the change of base formula for logarithms And it's a very simple observation how this works like if you take log base a of x and suppose This is equal to y switch it to the exponential form. This would tell us that x equals a to the y Then take the log base b of both sides of this equation So you get log base b of a to the y well on the right hand side the third law of Logarithms comes into play where you can pull an exponent out and put it as a coefficient So this becomes y times the log base b of a in the situation So what you're going to then do is you're going to divide both sides of the equation by log base log base b of A so that cancels out over here And so then what we see is you see that y is also equal to the log base b of x Over the log base a excuse me log base b of a so it's a it's a very quick It's a very quick Observation here that the third law gives us this change of base formula So how does one use the change of base formula now sadly the most common use for of it is something like the following approximate? log base 5 of 89 to 4 decimal places well You'll notice that there's no rational power of 5. That's going to give you 100 or give you 89 right? And so you console your calculator and you're searching you're searching But there's no there's no base 5 button for logarithms There's you'll probably see on your calculator a natural log button right those usually come equipped with a scientific or graphing calculator You might see one that just says log which will mean the common log right? But there's no log base 5 and so it's like what what do you do? And I swear it's like a conspiracy between like textbook companies and calculators that why couldn't why do we need both buttons? Well, you know, why don't we just have like a a log? You know a log b x button right where you type in two parameters b and x or something like that? If it's if the concern is you know space on the keyboard I think one button we get away from it you still have to have the common log You know excuse me the natural log you need so you make like a you need like a base e button Which probably is already there on the keyboard so Whatever but you know we live in these we live in primitive times where no one wants to put a universal log button on the calculator The change of base formula allows a convenient way to get around that because the log base 5 of 89 by the change of base formula This is equal to the log base b of 89 divided by the log base b of 5 for which that base can be any number you want you could switch it to the common log base 10 You could switch it to the natural log and so this is actually equal to the natural log of 89 over the natural log of 5 For which these are things you can compute on your calculator The calculator if you ask at the natural log of 89 this will be approximately 4.488 64 If you ask the natural log of 5 you'll get 1.609 4 4 And then if you take the ratio of those two numbers together again on your calculator screen You'll get approximately 2.7 8 8 9 of course rounding to four decimal places right there So that's how most students actually use the change of base formula as a way of just making up for the defect of the dumb calculator What were you thinking textures instruments? I don't know now. That's of course from a mathematicians perspective That's like the most useless reason to have a change of base formula one that could be easily fixed by a competent calculator factory Really the point of the change of base formula is how do you deal with log when you have different bases? Like what if you want to compute the expression log base 5 of 8 times log base 2 of 25, right? What if you have different bases? How do you work with these in harmony with one another? Well, honestly, you look at something like this. It's like log base 5 of 8 log base 5 of 25 Why couldn't you like switch the numbers around? Right? It would be a lot easier. I mean log base 2 of 8 That's pretty easy log base 5 of 25. That's super easy Turns out maybe you're not going to believe me here But by the change by the change of base formula you actually are justified in moving those bases around let me show you So we're going to take log base 5 of 8 and log base 2 of 25 I've used the change of base formula. I'm going to switch it to some other base and for lack of a better choice I'm going to switch it to the natural log. So this becomes the natural log of 8 over the natural log of 5 Right, so log base 5 of 8 becomes natural log of 8 over the natural log of 5 And we're going to find then also the log base 2 of 25 becomes the natural log of 25 over the natural log of 2 And Then what you can do is then you since this is multiplication you can swap locations could chink So you're going to get the natural log of 8 over the natural log of 2 And then you're going to times that by the natural log of 25 over the natural log of 5 Like so and now looking at the first fraction natural log of 8 over natural log of 2 That's the same thing as the log base 2 of 8 And then the second one the natural log of 25 over the natural log of 5 That's the natural log excuse me the log base 5 of 25 So what I said was actually correct. You were justified in swapping the bases in terms of multiplication. How cool is that? Now 2 of course what power of 2 gives you 8? That's the third power and what power of 5 gives you 25? That's the second power. So this product turns out to be 6 Something we can see of course when we use the change of base formula