We can use many bases for a logarithm, but the bases most typically used are the bases of the common logarithm and the natural logarithm. The common logarithm has base 10, and is represented on the calculator as log(x). The natural logarithm has base e, a famous irrational number, and is represented on the calculator by ln(x). The natural and common logarithm can be found throughout Algebra and Calculus.
Logs of a special bases so what we're going to talk about now is two different logs that have a base that arises enough that we have special notation for it. The first one is going to be what we call the common log which is just written log x. Now there is really a base there's no base written here but there are really is a base and it's going to be base 10, so whenever you see a log with no base with no little number down there, it's really going to be log 10 okay it's used enough that instead of having to write 10 every single time we write it we just leave it off we know that it is base 10. Okay so for instance, log 10 okay this is really log base there's a little invisible 10 right there log 10 of 10, 10 to the first is equal 10 so this is just going to be 1 okay? Log of 100, again there's little invisible 10 right here so it's saying 10 to a power is a 100 this is going to be 2. So just because there's no base doesn't, don't let that scare you just a little imaginary 10 is right there. They helps you to write it in, it never hurts but generally you're going to dealing with an enough that just sort of they use to having this be log base 10 kind of accepted and know what to do with it. Okay the other base that we're going to be dealing with is base e and this is going to sound a little weird okay? So natural this is called the natural log when there's no log is just ln okay and this is actually log base e of x and the easiest way understand e let's think of it sort of some word the pi you know you remember pi is a symbol 3.14, e is very similar except that e is a number 2.71 so on and so forth that's like pi it doesn't actually end it's just going to be a never ending decimal so pi is a number a little bigger by 3, e is a number a little bit less than 3 whenever you see ln know that we're dealing with log base e okay? So ln of e again this is just saying log base e of e when those numbers two agree this is always going to be 1, ln of e to the fourth you could use the laws of exponents and bring the 4 around here if you wanted to, so then you have 4 times lne 4 times 1 4, or just know that saying this is saying e to what power is equal to the e to the fourth just going to be 4. Okay, so two different notations log, natural log same rules applies everything else we've talked about just instead of having a base the base is part of the notation.