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Sal evaluates log₂(8), log₈(2), log₂(⅛), and log₈(½). Created by Sal Khan.
Video transcript
Let's give ourselves a little more practice with logarithms. So just a little bit of review... Let's evaluate log, base 2, of 8. What does this evaluate to? Well it's asking us it'll evaluate to the power that I have to raise our base to. That I have to raise 2 to, to get to 8. So 2 to the first power is 2, 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 did examples like that in the last video. Let's do something a little bit more interesting... What is-- and I'll colour code it... 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, 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. Well if 2 to the third power is 8, 8 to the one-third power is equal to 2. So 'x' is equal to one-third. 8 to the one-third power is equal to 2. Or you could say that the cube root of 8 is 2. So in this case 'x' is one-third. This logarithm right over here will evaluate to one-third. Fascinating. Let's mix it up a little bit more. Let's say we have the log, base 2, instead of 8 let's put a one-eighth, 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 one-eighth. So if we set this to be equal to 'x', we're essentially saying 2 to the 'x' power, is equal to one-eighth. Well we know that 2 to the third 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 one-eighth, which is a reciprocal of 8. We just have to raise to the negative 3 power. 2 to the negative 3 power is one over 2 to the third power. Which is the same thing as one over eight. So, if we're asking ourselves "what exponent do we have to raise 2 to to get to one-eighth?" 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, base 8, of one-half. What does this evaluate to? Let me clean this up so we have some space to work with. So as always, we're saying "what power do I have to raise 8 to, to get to one-half?" So let's think about that a little bit. We already know that 8 to the one-third power is equal to 2. If we want the reciprocal of 2 right over here, we have to just raise 8 to the negative one-third. So let me write that down, 8, to the negative one-third power, is going to be equal to one over eight, to the one-third power, and we already know the cube root of 8 or 8 to the one-third power is equal to 2. This is equal to one-half. So, the log, base 8, of one-half is equal to? Well the power I have to raise 8 to to get to one-half is negative one-third. I hope you enjoyed that as much as I did.