Let's simplify this
expression, or at least
let's try to write it
down as an integer using
my laws of logarithms.
Now, there are a
few ways to do this,
but let's use the laws
we've just learned.
So log to the base
3 9 to the 5--
well, that 5 is
inside the bracket,
so it's inside the parentheses.
It's being fed as part of
the input into the logarithm.
So this is 5 log
to the base 3 of 9.
So this is our log law we've
just seen, because 9 to the 5
is what we're taking the
logarithm of, the whole thing.
But now, log to the
base 3 of 9 is what?
That's going to be equal to--
well, 9 is 3 squared,
so 3 to the power 2.
So this has to be
equal to 5 times 2.
This is equal to 10.
Very simple.
So this is a useful thing
to do, because we basically
simplify the problem.
Instead of having to work
out, this is a power of 3--
which we can do, using laws of
exponents, a bit of a pain--
we can have completely traded
the problem for this one, which
is much more straightforward.
So this is a really useful way
that this kind of law will come
[? into ?] [? being, ?]
will be used.
So what about the
other expression?
Well, this is part
of the warning
I've given, that it's very
important to correctly
interpret what's going on.
So log to the base 3 of
9, all to the power 5.
Well, what does this mean?
Well, first of all, you
do log to the base 3 of 9.
Well, that's 2.
So it's 2 to the power 5.
That's 32.
And these are not
equal to each other.
So this, again, is
trying to let you
understand the importance of
writing things down carefully.
If you write this incorrectly--
I mean, something that people
write a lot is stuff like this.
They remove the bracket
there, and then it's
very ambiguous what you mean.
Do you mean the first thing?
Or do you mean-- well, exactly
what I've written here,
with the parentheses?
So make sure your
notation is correct.
Very, very important.
So always, whenever
you're taking
the logarithm of something,
put the entire thing
you're taking the logarithm
of within the parentheses
after the log to the given base.
