In this video, I am going to teach you how to do logarithms. Now, I have one handy trick for logarithms and it works throughout. First of all, we need to know all the basics of what the logarithm is. Now, first, we know that three squared is equal to nine, this means three times three, three squared, you write three two times, it's nine. Three times three times three, you've written 3 three times, that's equal to 27, so three cubed is equal to twenty seven. Now, a logarithm basically undoes this process. If I wish to write this, log 3, 9, this is to me, 3 to the power of what is equal to 9? We know 3 to the power of 2 is 9, so the answer is 2, so equivalently log 3, 27, you just have to say it to yourself, 3 to the power what is 27? We know three cubed or 3 to the power of 3 is 27, the answer, equivalently if we looked at log 2 of 16, this says, 2 to the power of what is 16. Now, how many times do we have to multiply 2 with itself to get 16? Now, 2 times 2, that's 4. 2*2*2 is 8, that's 2 cubed. 2*2*2*2 is 16. 2 to the power of what is 16, 2 to the power of 4 is sixteen, so the answer there is 4. That is the basics of logarithms. Some identities we're going to cover about logarithms, we know for sure x to the power 0 is 1, no matter what x is, it can be any number, x to the power 0 is 1. So, what is this then? Log x 1, this is x to the power what is 1? The answer has to be 0. Because x to the power 0 is equal to 1, and using that term x to the power what is 1, will give you the answer 0 every time. Okay, so a couple of identities with logarithms, we have this, logarithm of xy is the logarithm of x multiplied by y, using the identity is log x + log y. Okay? When they are multiplied within the bracket, they add outside the bracket. Similarly, with divide, you have x/y. Obviously, we have log x-log y. There are a couple of useful logarithms that will help you throughout. There's one more, log of x to the power d is equal to d log x, that one is exceedingly useful. Okay, so, finally, there is one proper logarithm that everyone uses, it's very common, and it relates to the natural number e. This natural number is simply like pi. It's just a number, it's equal to 2.71 something. It's a bit like pi, but it's used a lot with logarithms. Now, the way we write log base e, say 10, instead of writing log bas e, we write in. That actually means natural logarithm of 10, and that's it, that's all you need to know. It's exactly the same principles, this means e to the power what is equal to 10? I can't do that on the top of my head, but you put into a calculator, and you can work it out. So, this here, it's called lon, and it's the natural logarithm of 10. And that's how we use logarithms. .

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