All right, so let's try
and solve this problem.
We want to find
all x such that 4
to the power of the
quadratic expression x
squared plus 1 is equal to 8.
So the exponent is varying here.
It makes sense to try to
apply logarithm laws here,
or logarithms to try
to undo this exponent.
And I think we should use log to
the base 4 for obvious reasons,
because the base here is 4.
So let's take this expression.
So if this is true, this is
if and only if I take logs--
preserves inequality or
preserves equality, of course.
This is log to the base 4 of 8.
OK, so now what's this?
This is if and only if.
By definition of a
logarithm really,
or maybe a
cancellation property,
this is just x squared plus 1.
And I'm left with log
to the base 4 of 8.
So let's continue.
Let's not worry too much
about log to the base 4 of 8
for a second.
It's correct, right?
It's not wrong.
We could maybe try
and simplify it.
But let's just push
through to see exactly
what happens before we do this.
So this is if and only if.
Of course, x squared is equal to
log to the base 4 of 8 minus 1.
And then of course, at the end,
thankfully, 4 to the power of 1
is 4.
So log to the base
4 of 8 is actually
going to be bigger than 1.
So this is positive, so
nothing weird happens.
It's going to be plus minus the
square root of this expression.
Now, you can simplify log to the
base 4 of 8 a bit if you want.
It's not completely necessary.
I'll do it now just
for completeness.
We've got the right answer.
So what is log to
the base 4 of 8?
Well, it's a bit annoying.
I mean, 8 is somewhere
between 4 to the power 1 and 4
to the power 2.
So it's between 4
and it's between 16.
So what am I going to do?
It's a bit confusing.
But notice that
log to the base 4,
it's log to the
base 4 of 2 times 4.
OK, of course.
And I know my laws
of logarithms tell me
that this has to
break apart as a sum.
So this has to be equal
to log to the base 4 of 2
plus log to the base 4 of 4.
And the thing now is 2
is the square root of 4.
So 4 to the power of
1/2 is equal to 2.
So the first part
is equal to 1/2.
And the second part was
just 1 by definition.
So just to make sure
we really understand
this specific step here, 4 to
the power of 1/2 is equal to 2.
So what does it
all come out to be?
It's 3/2.
So you can simplify this
quite a lot if you want.
But there's no real need.
It's just an extra
cherry on top.
You've got the answer.
This is correct, right?
You don't need to continue.
I know that very
often in high school,
you're given very
specific instructions
that everything has been put
in at exactly the right format.
That's just ridiculous.
It doesn't need to be.
As long as you have the
right answer, it's fine.
The only reason you
might potentially
have to continue
from this is if you
needed to use it
for something else,
and you maybe would need to
have it in a nice format.
But the answer is
correct as it stands.
