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# Logarithmic equations: variable in the base

Sal solves the equation 4=log_b(81). Created by Monterey Institute for Technology and Education.

Video transcript

Solve 4 is equal to
log base b of 81 for b. So let's just remind ourselves
what this equation is saying. This is saying, if I raise
b to the fourth power, then I'm going to get 81. Let me rewrite that. So if I raise b, that's our
base, so if I raise our base, b-- I guess that's why
they picked b. b for base. If I raise b to
the fourth power-- I'll do the fourth
power in orange. If I raise b to
the fourth power, that is going to be equal to 81. I've just rewritten
this equation, this logarithmic equation,
I've rewritten it as an exponential equation. So it says b to the fourth
power is equal to 81, and we need to
still solve for b. So you just have to
think, what number do I have to multiply by itself
four times in order to get 81. And 81 might jump out of
you, it is a perfect square. We know that 9 times
9 is equal to 81. Or we know that
another way to say that is, nine squared
is equal to 81. But we have to raise
something to the fourth power. But 9 itself is 3 times 3. So one 9 is 3 times 3, and then
you multiply it by another 9. That's another 3 times 3. That will also be equal to 81. And we can verify it. 3 times 3 is 9, 9 times 3
is 27, 27 times 3 is 81. So this is 3 to
the fourth power. 9 squared is the same thing
as 3 to the fourth power. So there we've done it. We've said, well, some
number to the fourth power is equal to 81. We know that 3 to the
fourth power is equal to 81. So we know that b is equal to 3. 3 to the fourth power is 81. Or we could say, log
base three of 81, this is saying what power do I
have to raise 3 to, to get 81? Well, we know. You have to raise 3 to the
fourth power to get to 81. And if you know about
fractional exponents, and don't worry about this if
this confuses you a little bit, you could raise both
of these, if you know about fractional exponents
and exponent properties, you could do it
this way as well. You could take, so you
have b to the fourth power is equal to 81. You could raise both of
these to the 1/4 power. Anything you do to
one side of equation, you have to do to
the other side. And from our
exponent properties, you know that if you raise
something to a power, then raise that
to a power, that's like raising it to
the 4 times 1/4 power. Or this is, essentially, just
raising it to the first power. So on the left hand side,
you're just left with b, and on the right hand side,
you're left-- is equal to 81 to the 1/4 power. But figuring out what
81 to the 1/4 power, you really have to go
through this exercise anyway. Because when you raise
something to the 1/4 power, you're really saying, well,
what do I have to multiply? What do I have to raise to
the fourth power to get to 81? And then you get what
81 is to the 1/4 power. So actually, this is
another way of realizing that 81 to the 1/4
power is equal to 3. 3 to the fourth
power is equal to 81. 81 to the 1/4 power
is equal to 3. But if this is confusing to you,
don't worry about it for now. The important thing
is you understand what the logarithm
is actually saying. If I raise b to the
fourth power I get 81.