The next property of logarithms has to do with exponents, or powers. When we
have log base b of x to the a, we can really set that equal to a times log base
b of x. In other words, we just take this exponent of a and we bring it down in
front of our logarithm as a coefficient. In other words, if I had log of base
10 of a 100 to the 3rd power, we can rewrite this. We can bring this 3rd power
down in front of our logarithm, so we'll have 3 times the logarithm of base 10,
of 100. Now I usually put this in parentheses, so I understand that it's 3
times this quantity. And let's carry out the map on both sides of the equation
to make sure that this makes sense. On the left hand side we'll have log of
base 10 and then 100 to the 3rd power. Well 100 to the 3rd power is the same
thing as a million. That's 1 followed by 6 zeroes. On the left hand side we'll
have 3 times the log base ten of 100. Log base 10 of 100 is 2. We know this,
since 10 squared equals 100. We're looking for the exponent that we'd raise 10
to, in order to get this number. So, we know 3 times 2 equals 6. And now let's
check the left-hand side. We have log base 10 of a million. But we know a
million is 1 followed by 6 zeros. So, that means we have to raise 10 to the 6th
power in order to get a million. So yes, 6 equals 6. This is just one example
of why this log property holds true. And finally, here's the last logarithm
property we'll examine. If we subtract two logarithms that have the same base,
then we can just divide the two numbers. So the log with base b of x minus the
log of base b of y equals the log of base b of x divided by y. For example, the
log of base 3 of 81, minus the log of base 3 of 9 would equal the log of base 3
of 81 divided by 9. We simply just divide these 2 numbers inside of our
argument and take the log of it. And let's also be sure that this checks. We
know the log of base 3 of 81 equals 4 since 3 to the 4th equals 81. We also
know the log of base 3 of 9 equals 2, since 3 squared equals 9. And, finally,
on the right-hand side, we'll have the log of base 3 of 81 divided by 9. We
know 81 divided by 9 really equals 9, so we have log of base 3 of 9. Now we can
just subtract here, so we get 2 equal to 2. We know log of base 3 of 9 is
really just 2, as before. Now, using those log properties, I want you to try
and write this expression with 1 logarithm. In other words, you should have the
log with base b of 1 number here. Now it's okay if you're not entirely sure
what to do, I just want to try your best. Think back to what it means when you
add logs together and when you subtract logs together. Do them one at a time
and I think you might get the answer. And again it's totally okay if you don't
get there. You can always see the solution.
