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# Evaluating logarithms (advanced)

Sal evaluates log₂(8), log₈(2), log₂(⅛), and log₈(½). Created by Sal Khan.

Video transcript

Let's give ourselves a
little bit more practice with logarithms. So just as a little bit of
review, let's evaluate log base 2 of 8. What does this evaluate to? Well, it's asking us
or it will evaluate to the power or
the exponent 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. And I'll color-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 1/3. 8 to the one-third
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 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 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, we're essentially saying 2 to
the x power is equal to 1/8. Well, we know 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 1/8,
which is the reciprocal of 8, we just have to raise 2
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 1/8. 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 base 8 of 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 do I 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
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 1
over 8 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 1/2. So the log base 8 of 1/2 is
equal to-- well, what power do I have to raise 8 to to
get to 1/2-- is negative 1/3. Equal to negative 1/3. I hope you enjoyed
that as much as I did.