>> Here we are going to
use the laws of logarithms
to evaluate several expressions.
The first one is log base 3
of 81 minus log base 3 of 3,
and we are to, first of
all, combine the expressions
into a single logarithm.
We have the difference of
logarithms with the same base.
That's going to be a
logarithm with that base
of the quotient of
those two values.
So log base 3 of, 81 divided by
3, which is log base 3 of 27,
which is log base 3 of 3 cubed.
And so the answer is 3.
The second example, log base
2 of 10, minus log base 2
of 35, plus log base 2 of 112.
So, in this case, we
work from left to right
and where we have the
difference of log base 2's,
we're going to write log base 2
of the quotient of
those arguments.
Then we will have plus
log base 2 of 112.
Now we're going to write
the sum of the log base 2
as the log base 2 of the
product of those arguments.
I see that 7 goes into
35 and also into 112,
and then 5 can cancel
into the 10.
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So we get log base 2 of
32.. 32 is 2 to the 5th..
and a log base 2 of 2 to the
5th is the exponent or 5.
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In the next example.. again..
we work from left to right.
Again, we'll have the difference
of the logs is the log
with the same base
of the quotient..
This time.. minus the log base
5 of 20, which will be the log
of the quotient again.
So 16 divided by 20,
all divided by 20.. which I'm
going to write as 20 over 1;
which will give log base 5 of
16 over 20 times 1 over 20.
Four goes into the 16, four
times, into the 20, five times.
And then again, 4 goes
into 4, once, and into 20, 5 times.
So we're getting log base 5 of
1 over 25. Which is log base 5
of the reciprocal of, 5 squared;
which is 5 to the negative 2.
And so the result is negative 2.
And the last expression
in the set:
The log of, the log of, 10 to
100,000 power.
So this time I can replace
the log of 10 to the 100,000
with 100,000 because it
is the exponent on a 10.
That leaves me with log of a
100,000, which is 10 to the 5th.
And that will leave me
with 5, since we're looking
at the common log, log
base 10 by default.
