Here is the definition of the logarithm function.
If b is any number such that b > 0
and b ≠1 and x > 0 then,
y = logb x is equivalent to
We usually read this as “log base b of x”.
In this
definition logb y = x is called the logarithm form and by = x is called the exponential form.
Note that the
requirement that x > 0 is really a
result of the fact that we are also requiring b > 0. If
you think about it, it will make sense. We are raising a positive number to an
exponent and so there is no way that the result can possible be anything other
than another positive number. It is very important to remember that we can’t
take the logarithm of zero or a negative number.
Now, let’s
address the notation used here as that is usually the biggest hurdle that
students need to overcome before starting to understand logarithms. First, the
“log” part of the function is simply three letters that are used to denote the
fact that we are dealing with a logarithm. They are not variables and they
aren’t signifying multiplication. They are just there to tell us we are dealing
with a logarithm.
Next, the b that
is subscripted on the “log” part is there to tell us what the base is as this
is an important piece of information. Also, despite what it might look like
there is no exponentiation in the logarithm form above. It might look like
we’ve got bx in that form, but it isn’t. It just looks like that might
be what’s happening.
It is important
to keep the notation with logarithms straight, if you don’t you will find it
very difficult to understand them and to work with them. Hopefully, you
now have an idea on how to evaluate logarithms and are starting to get a grasp
on the notation. There are a few more evaluations that we want to do however,
we need to introduce some special logarithms that occur on a very regular
basis. They are the common logarithm and the natural logarithm.
Here are the definitions and notations that we will be using for these two logarithms.
common logarithm : log
x = log10 x
natural logarithm : ln
x = loge x
So, the common logarithm
is simply the log base 10, except we drop the “base 10” part of the notation.
Similarly, the natural logarithm is simply the log base e with a
different notation and where e is the same
number that we saw in the previous section and is defined to be e = 2.718281827….
Properties of
Logarithms
1. logb 1 = 0 .
This follows from the fact that b0 =1.
2. logb b = 1 .
This follows from the fact that b1 = b .
3. logb bx = x . This can be generalize out to logb bf (x) = f (x) .
Properties
3 and 4 leads to a nice relationship between the logarithm and exponential
function.
Let’s
first compute the following function compositions for f (x) = bx
and g(x) = x .
Recall
from the section on inverse
functions that this means that the exponential and
logarithm functions are inverses of each other. This is a nice fact to remember
on occasion.
We
should also give the generalized version of Properties 3 and 4 in terms of both
the natural and
common logarithm as we’ll be seeing those in the next couple of sections on
occasion.
ln e f (x) = f (x)
e ln f
(x)
= f (x)
Now,
let’s take a look at some manipulation properties of the logarithm.
More
Properties of Logarithms
For these properties we will assume that x > 0.
5. logb (xy) = logb
x + logb y
6. logb (x/y) = logb
x – logb y
7. logb (xr)
= r logb x
8. If logb x
= log y then x = y.
We won’t be
doing anything with the final property in this section; it is here only for the
sake of completeness. We will be looking at this property in detail in a couple
of sections.
The first two
properties listed here can be a little confusing at first since on one side
we’ve got a product or a quotient inside the logarithm and on the other side
we’ve got a sum or difference of two logarithms. We will just need to be
careful with these properties and make sure to use them correctly.
Also, note that
there are no rules on how to break up the logarithm of the sum or difference of two terms. To be clear about this let’s note the following,
log (x + y) ≠ logb x + logb
y
log (x – y) ≠ logb
x – logb y
Be careful with these and do not try to use these as they simply
aren’t true.
Note that all
of the properties given to this point are valid for both the common and natural logarithms. We
just didn’t write them out explicitly using the notation for these two
logarithms, the properties do hold for them nonetheless.
Now, let’s see some examples of how to use these properties.
Example 1 Simplify each
of the following logarithms.
Solution
The
instructions here may be a little misleading. When we say simplify we really
mean to say that we want to use as many of the logarithm properties as we can.
Note that we
can’t use Property 7 to bring the 3 and the 5 down into the front of the
logarithm at this point. In order to use Property 7 the whole term in the
logarithm needs to be raised to the power. In this case the two exponents are
only on individual terms in the logarithm and so Property 7 can’t be used here.
We do, however,
have a product inside the logarithm so we can use Property 5 on this logarithm.
Now that we’ve
done this we can use Property 7 on each of these individual logarithms to get
the final simplified answer.
In this case
we’ve got a product and a quotient in the logarithm. In these cases it is
almost always best to deal with the quotient before dealing with the product.
Here is the first step in this part.
Now, we’ll
break up the product in the first term and once we’ve done that we’ll take care
of the exponents on the terms.
For this part
let’s first rewrite the logarithm a little so that we can see the first step.
Written in this
form we can see that there is a single exponent on the whole term and so we’ll
take care of that first.
Now, we will
take care of the product.
Notice the parenthesis
in this the answer. The 1 / 2 multiplies the original logarithm and so it will also
need to multiply the whole “simplified” logarithm. Therefore, we need to have a
set of parenthesis there to make sure
that this is taken care of correctly.
We’ll first
take care of the quotient in this logarithm.
We now reach
the real point to this problem. The second logarithm is as simplified as we can make it.
Remember that we can’t break up a log of a sum or difference and so this can’t
be broken up any
farther. Also, we can only deal with exponents if the term as a whole is raised
to the exponent. The fact that both pieces of this term are squared doesn’t
matter. It needs to be the whole term squared, as in the first logarithm.
So, we can
further simplify the first logarithm, but the second logarithm can’t be
simplified any more. Here is the final answer for this problem.
.
The final topic
that we need to discuss in this section is the change of base formula.
Most
calculators these days are capable of evaluating common logarithms and natural
logarithms. However, that is about it, so what do we do if we need to evaluate another
logarithm that can’t be done easily as we did in the first set of examples that
we looked at?
To do this we have the change of base formula. Here is the change
of base formula.
where we can
choose b to be anything we want it to be. In order to use this to help
us evaluate logarithms this
is usually the common or natural logarithm. Here is the change of base formula using
both the common logarithm and the natural logarithm.
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