Let's talk about calculating logarithms. So the first way is to just use the fact
that a logarithm is a way to explain
exponents. If you see something like log
base 3 of 9 equals what, a way to
understand logarithms is to convert them
to exponents. So this would be 3 to what
power is 9. Well, 3 squared is 9, so that
means log base 3 of 9 is 2. If you
think of a log in terms of an
exponential function, then sometimes you
can easily figure out what this number
is. Similar idea here. If I think of this
as an exponential, I'll have 2 to what
number gives me 8. Well, 2 cubed is 8, so
that means that log base 2 of 8 equals 3.
That's not going to happen in class most of
the time. Most of the time you're going to
have crazy examples like this: 3 to what
power equals 5? Well, 3 to the 1 power is
3. 3 squared is 9. So how do I get a 5 out
of that? It's somewhere between 1 and 2. I
don't want to have to guess and check.
that is where the
change your base formula comes into
play. If you have log base a of X, you can
type this into your calculator using a
log button or a natural log button. This
is the common log. Everyone should have a LOG button on your calculator. This is
the natural log. Everyone should have a
LN button on your calculator.
You can just pick which one of these
fractions you want to use. So you write
common log of x over common log of a or
natural log of x over natural log of a.
It'll give you the same answer. Let's do
those same two examples that we had
earlier. That means that I want to calculate
something like log base 3 of 5. I need to
use change of base formula, and I'm going
to write log over log.
I always remember which one to put on
top of my fraction thinking through the
5 is physically higher up than the 3, so
the 5 needs to be higher in my fraction.
Alright, let's type this into your
calculator. Log 5 - close your parentheses -
divided by log 3. That means this is one
point four six. If we wanted to go back
to when we had converted it, you can
always double check your answer. 3 to the
1.46 power should give you
something close to 5. It's a little bit
off because 1.46 is rounded. If
I didn't want to use common logs there, I
could have used natural logs - natural log
of 5 over natural log of 3 - and it would
have given me the same answer. Alright, let's do one more example. If I wanted
to calculate this using my calculator,
it's going to be log over log or natural
log over natural log. The 3 is physically
higher up than the subscript 7, so that
reminds me it'll be 3 over 7. Log 3
divided by log 7 tells me that log base
7 of 3 is 0.56. If you wanted to use
natural logs instead of common logs, it
will give you the same answer. Natural
log 3 divided by natural log of 7. It
gives you the same answer. And some of
you will be lucky enough to have an
updated calculator where you can
actually calculate these without having
to use change of base. If you are one of
those lucky people, you need to press
the math button and scroll all the way
down or up to A:logBASE. In that
case, I can type in my little seven. I can
type in my three, and it will give me the
same exact answer that I got using
change of base formula. So again, that was
the math button scroll all the way down
to A, the letter A:logBASE.
