�Introduction To Logarithms
Logarithms were originally
developed to simplify complex
arithmetic calculations.
They were designed to transform
multiplicative processes
into additive ones.
If at first this seems like no big deal,
then try multiplying
2,234,459,912 and 3,456,234,459.
Without a calculator !
Clearly, it is a lot easier to add
these two numbers.
Today of course we have calculators
and scientific notation to deal with such
large numbers.
So at first glance, it would seem that
logarithms have become obsolete.
Indeed, they would be obsolete except for one
very important property of logarithms.
It is called
the power property and we
will learn about it in another lesson.
For now we need only to observe that
it is an extremely important part
of solving exponential equations.
Our first job is to try to make some sense of logarithms.
Our first question then must be:
What is a logarithm ?
Of course logarithms have
a precise mathematical
definition just like all terms in mathematics. So let’s
start with that.
Definition of Logarithm
Suppose b>0 and b≠1,
there is a number ‘p’
such that:
logb n = p if and only if bp = n
Now a mathematician understands exactly what that means.
But, many a student is left scratching their head.
The first, and perhaps the most important step, in understanding logarithms is to realize that they always relate back to exponential equations.
You must be able to convert an exponential equation into logarithmic form and vice versa.
So let’s get a lot of practice with this !
Example 1:
Solution:
We read this as: ”the log base 2 of 8 is equal to 3”.
log28 = 3
Example 1a:
Solution:
Read as: “the log base 4 of 16 is equal to 2”.
log416 = 2
Example 1b:
Solution:
log2(1/8) = -3
Okay, now it’s time for you to try some on your own.
Solution:
log51= 0
Solution:
Solution:
It is also very important to be able to start with a logarithmic expression and change this into exponential form.
This is simply the reverse of
what we just did.
Example 1:
Solution:
Example 2:
Solution:
Okay, now you try these next three.
Solution:
Solution:
Solution:
We now know that a logarithm is perhaps best understood
as being
closely related to an
exponential equation.
In fact, whenever we get stuck
in the problems that follow
we will return to
this one simple insight.
We might even state a
simple rule.
When working with logarithms,
if ever you get “stuck”, try
rewriting the problem in
exponential form.
Conversely, when working
with exponential expressions,
if ever you get “stuck”, try
rewriting the problem
in logarithmic form.
Let’s see if this simple rule
can help us solve some
of the following problems.
Solution:
Let’s rewrite the problem in exponential form.
We’re finished !
Solution:
Rewrite the problem in exponential form.
Example 3
Try setting this up like this:
Solution:
Now rewrite in exponential form.
Our final concern then is to determine why logarithms like the one below are undefined.
Can you give an explanation?
One easy explanation is to simply rewrite this logarithm in exponential form.
We’ll then see why a negative value is not permitted.
First, we write the problem with a variable.
Now take it out of the logarithmic form
and write it in exponential form.
What power of 2 would gives us -8 ?
Hence expressions of this type are undefined.
That concludes our introduction
to logarithms. In the lessons to
follow we will learn some important
properties of logarithms.
One of these properties will give
us a very important tool
which
we need to solve exponential
equations.