Evaluating logarithms (advanced)

let's give ourselves a little bit more practice with logarithms so just as a little bit of review let's evaluate log base two of eight what does this evaluate to well it's asking us or they will evaluate to the power that I have toward the exponent that I have to raise our base two that I have to raise 2 to to get to 8 so 2 to the first power is 2 to the second power is 4 2 to the third power is 8 so this right over here is going to be equal to 3 fair enough we need examples like that in the last video let's do something a little bit more interesting what is what is an all color-coded what is log base 8 what is log base 8 of 2 now this is interesting I'll give you a few seconds to think about it well we're asking ourselves or this will evaluate to the exponent that I have to raise 8 to to get to 2 so let's think about that in another way so we could say 8 8 to some power to some power and that exponent that I'm raising 8 to is essentially what this logarithm would evaluate to 8 to some power is going to be equal to 2 is going to be equal to 2 well if 2 to the third power is 8 8 to the 1/3 power is equal to 2 so X is equal to 1/3 8 to the 1/3 power is equal to 2 or you could say the cube root of 8 is 2 so in this case X is 1/3 this logarithm right over here will evaluate 2 will evaluate to 1/3 fascinating let's mix it up a little bit more let's say we had the log base 2 instead of 8 let's put a let's put a 1/8 1/8 right over here so I'll give you a few seconds to think about that well it's asking us or this will evaluate to the exponent that I have to raise 2 to to get to 1/8 so if we said that this is equal to X were essentially saying two to the X power 2 to the X power is equal to 1/8 is equal to 1/8 well we know 2 to the 3rd power let me write this down we already know that 2 to the third power is equal to 8 if we want to get to 1/8 which is the reciprocal of 8 we just have 2 raised to the negative 3 power 2 to the negative 3 power is 1 over 2 to the third power which is the same thing as 1 over 8 so if we're asking ourselves what exponent do we have to raise 2 to to get to 1/8 well we have to raise it to the negative 3 power so X is equal to negative 3 this logarithm evaluates to negative 3 now let's really really mix it up what would be the log what would be the log base 8 base 8 of 1 1/2 what does this evaluate to let me clean this up so that we have some space to work with so as always we're saying what power to have to raise 8 to to get to 1/2 so let's think about that a little bit we already know that 8 to the 1/3 power is equal to 2 if we want the reciprocal of 2 right over here we have to just raise 8 to the negative 1/3 so let me write that down 8 to the negative 1/3 power is going to be equal to 1 over 8 to the 1/3 power and we already know the cube root of 8 or 8 to the 1/3 power is equal to 2 this is equal to 1/2 so the lot log base 8 of 1/2 is equal to well what power after raise 8 to to get to 1/2 is negative 1/3 it's called a negative 1/3 I hope you enjoyed that as much as I did