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# Proof of the logarithm change of base rule

## Video transcript

what I want to do in this video is prove the change of base formula for logarithms change of base change of base formula which tells us write this formula formula which tells us that if I want to figure out the logarithm base a base a of X base a of X that I can figure this out by taking logarithms with a different base that this could be that this would be equal to the logarithm logarithm base B so some other base base B of X base B of X divided by divided by the logarithm base B base B of a base B of A and this is a really useful result if your calculator only has natural logarithm or log base ten you can now use this to figure out the logarithm using any base if you want to figure out the log base two let me make it clear if you want to figure out the logarithm base let's say base three of let's say 25 you can use your calculator either using log base 10 or log base 2 so you could say that this is going to be equal to log base 10 log base 10 of 25 and most calculators have a button for that divided by divided by log base 10 divided by log base 10 of 3 log base 10 of 3 so this is an application of the change of base formula but let's actually prove it so let's say that let's say that we want it let's let's set logarithm logarithm base a of X base a of X to be equal to some new variable let's call that variable let's call that equal to Y so this right over here we are just setting that equal to Y well this is just another way of saying that a to the Y power is to X so we can rewrite this as a to the Y power a to the Y power is equal to is equal to X and I'll write the X out here because I'm about to these two things are equal this is just another way of restating what we just wrote up here now let's introduce the logarithm base B and to introduce it I'm just gonna take log base B of both sides of this equation so let's take a logarithm base B of the left-hand side and a logarithm base B logarithm base B of the right-hand side well we know from our logarithm properties that the logarithm of something to a power is the exact same thing as the power times the logarithm of that something so logarithm base B of a to the Y is the same thing as Y times the logarithm the logarithm base B of a so this is just a traditional logarithm property we proved it elsewhere and we already know it's going to be equal to the right hand side it's going to be equal to log base B log base B of X and now let's just solve for y and this is exciting because why was this thing right over here but now if we solve for y we're gonna be solving for y in terms of logarithm base B to solve for y we just have to divide both sides of this both sides of this equation by log base B of A so we divide by log base B of a of a on the left hand side and we divide by log base B base B of a on the right hand side and so on the left hand side these two characters are going to cancel out and we are left with and we deserve a drum roll now that Y y is equal to log base B of X divided by log base B of A so let me write it just copy and paste this so I don't have to keep switching colors so let me paste this so there you have it you have your change of base formula remember Y is the same thing this thing right over here why is log of a actually let me make it clear why which is equal to log of a which is equal to log base a of X so copy and paste Y which is equal to this thing from which is how we defined it right over here Y is equal to log base a of X is we've just shown is also equal to this if we write it in terms of base if we write it in terms of base B and we have our change of base formula