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Evaluating logarithms: change of base rule

Video transcript

use the change of base formula to find log base 5 of a hundred to the nearest thousandth so the change of base formula is a useful form especially when you're going to use a calculator because most calculators don't aren't don't allow you to arbitrarily change the base of your logarithm they have functions for log base E which is a natural logarithm and log base ten so you generally have to change your base and that's what the change of base formula is and if we have time I'll tell you why it makes a lot of sense or how we can derive it so the change of base formula just tells us that log and let me do some colors here log base a of B is the same thing is the exact same thing as as log base X where X is an arbitrary base of B of B over over log base that same base base X base X over a over a and the reason why this is useful is that we can change our base here our bases a and we can change it to base X so if our calculator has a certain base X function we can convert to that base it's usually e or base 10 base 10 is an easy way to go and in general if you just see someone write a logarithm like this if they just write log of X they're implying this implies log base 10 of X if someone writes natural log of X they are implying log base e of X and E is obviously the number two point seven one keeps going on and on and on forever now let's apply it to this problem we have we need to figure out the logarithm I'll use the colors base five base five of 100 of 100 so this property this change of base formula tells us this is the exact same thing as log I'll make X 10 log base 10 of 100 of 100 divided by divided by log base 10 of 5 of 5 and actually we don't even need a calculator to evaluate this top part log base 10 of 100 what power to have to raise 10 to to get to 100 the second power so this numerator is just equal to 2 so it simplifies to 2 over log base 10 of 5 and we can now use our calculator because the log function on a calculator is log base 10 so let's get our calculator out and we're going to get we want let me clear this 2 divided by when someone just writes log their mean they mean base 10 they press Ln that means base e so log without any other information is log base 10 so this log base 10 of 5 is equal to is equal to 2 point and they want us to the nearest thousandth so two point eight six one so this is approximately equal to two point eight six one and we can verify it because in theory if I raise 5 to this power I should get one hundred and it kind of makes sense because 5 to the second power is 25 5 to the third power is 125 and this is in between the two and it's closer to the third power than it is to the second power and this number is closer to 3 than it is to two but let's verify it let's verify it so if I take five to that power if I take five and let me type in well let me just type in what we did to the nearest thousandth five to the 2.86 one so I'm not putting in all of the digits what do I get I get ninety-nine point nine four if I put all of these digits in it should get pretty close to 100 so that's what makes you feel good that this is the power that I have to raise 5 to to get to 100 now with that out of the way let's actually think about why this property why this thing right over here makes sense so if I write if I write log base a I'm trying to be fair to the colors log base a of B let's say I set that to be equal to let's say I set that equal to some number let's call that equal to C or I could call it e first well I'll say that's equal to C so that means that means that a to the C power that means that a to the C power a to the C power is equal to B these this is an exponential way of writing this truth this is a logarithmic way of writing this truth is equal to B now we can take the logarithm of any base of both sides of this anything you do if there's if you say 10 to the what power equals this 10 to the same power will be equal to this because these two things are equal to each other so let's let's take the same logarithm of both sides it's the logarithm with the same base and I'll actually do log base X to prove the general case here so I'm gonna take log base X of both sides of this so this is log base X of a to the C power a to the C power I try to be faithful to the colors is equal to is equal to log base X of B is equal to B is equal to B and let me close it off in orange as well and we know from our logarithm properties log of a to the C is the same thing as C times the logarithm of whatever base we are of a logarithm of a logarithm of a and of course this is going to be equal to log base X of B log base X of let me put it like I just write a B right over there and if we wanted to solve for C you just divide both sides by log base X of a so you get C C is equal to and I'll stick to the color it's just log base X of B which is this over log log base X of a and this is what C was C was log base a of B it's equal to log base a be let me write it this let me write it she let me well let me let me do the original color codes just so it becomes very clear what I'm doing I think you know where this is going but I I want to be fair to the colors so C is equal to log base X of B of B over over let me scroll down a little bit log base X dividing both sides by that of a of a and we know from here I can just copy and paste it this is also equal to C this is how we defined it so let me copy it and then let me paste it so this is also equal to C and we're done we've proven we've proven the change of base formula log base a of B is equal to log base X of B divided by log base X of a in this example a was 5 B is 100 and the base that we switched it to is 10 X is 10