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CCSS.Math:

let's learn a little bit about the wonderful world of logarithms so we already know how to take exponents if I were to say 2 to the 4th power what does that mean well that means 2 times 2 times 2 times 2 2 multiplied or repeatedly multiplied 4 times and so this is going to be 2 times 2 is 4 times 2 is 8 times 2 is 16 but what if we think about things in another way what if we're essentially we know that we get to 16 when we raise 2 to some power and we want to know what that power is so for example let's say that I start with 2 and I say I'm raising it to some power what does that power have to be to get 16 well we just figured that out you would X would have to be X X would have to be 4 and this is what logarithms are fundamentally about figuring out what power you have to raise to to get another number now the way that we would denote this with logarithm notation is we would say log base actually let me make that let me make it a little bit more colorful log base 2 logs on this 2 in blue log base 2 of 16 of 16 of 16 is equal to what or is equal in this case since we have the X there is equal to X this and this are completely equivalent statements this is saying hey well if I take 2 to some x power I get 16 this is saying what power do I need to raise 2 to to get 16 and I'm going to set that to be equal to X and you would say well you got to raise it to the fourth power once again X is equal to 4 so with that out of the way let's try more examples of evaluating logarithmic expressions so let's say you had let's say you had log base 3 log base 3 of 81 what would this evaluate to well this is a reminder this evaluates to the power we have to raise 3 to to get to 81 so if you want to you could set this to be equal to an X set that to be equal to the next and you can restate this equation as 3 as 3 to the X power 3 to the X power is equal to 81 why is a logarithm useful and you'll see that it has very interesting properties later on but you didn't necessarily have to use algebra to do it this way to say that the x is the power that you raise 3 to to get to 81 you had to use algebra here well with the just a straight up logarithmic expression you didn't really have to use in the algebra we didn't have to set it equal to X we could just say this evaluates to the power I need to raise 3 to to get to 81 the power I need to raise 3 to to get to 81 well what power do you have to raise 3 to to get to 81 well let's experiment a little bit so 3 to the first power is just 3 3 to the second power is 9 3 to the third power is 27 3 to the fourth power 27 times 3 is equal to 81 3 to the fourth power is equal to 81 X is equal to 4 so we could say log base 3 base 3 of 81 of 81 is equal to I'll do this in the new color I haven't have used almost every color is equal to 4 let's do several more of these examples and I really encourage you to give a shot on your own now that we get and you'll hopefully get the hang of it so let's try some a little larger number let's say we want to take log base 6 of 216 what will this evaluate to well we're asking ourselves what power do we have to raise 6 to to get to 216 6 to the first power 6 6 to the second power is 36 thirty-six times six is 216 this is equal to 216 so this is six to the third power is equal to 216 so if someone says what power do have to raise six to this base here to get to 216 well that's just going to be equal to three six to the third power is equal to 216 let's try another one let's say I had I don't know log base 2 of 64 so what does this evaluate to well once again we are asking ourselves so this will evaluate to the exponent that I have to raise this base two and you do this is this little subscript right here the exponent that after erase to two to get to 64 so 2 to the first power is 2 to the second power is 4 8 16 32 64 so this right over here is 2 to the sixth power is equal to 64 so when you evaluate this expression you say what power to have to raise 2 to to get to 64 well I have to raise it I have to raise it to the 6th power let's do it slightly let's do a slightly more straightforward one or maybe this will be less straightforward depending on how you view it what is log base 100 of 1 let you think about that for a second and this one so the hundred is the subscript of them.the it's log base 100 of 1 that's one way to think about I could put a parenthesis around the 1 what does this evaluate - well this is asking ourselves or we would evaluate this as what power to have to raise 10 to today sorry what part enough to raise a hundred - to get to 1 so let me write this down as an equation so if I set this to be equal to X this is literally saying hundred - what power is equal to 1 well anything that the zeroth power is equal to 1 so in this case X is equal to 0 so log base 100 of 1 is going to be equal to zero log base anything of one is going to be equal to zero because anything to the zeroth power and we're not talking about zero here anything to the zeroth power is that's not zero is going to be equal to one