Natural Logs

Date: 12/20/97 at 12:56:18
From: Alison
Subject: Natural logs

Hello, 

I have tried for the past two weeks to find what I could on logs, and 
mostly natural logs, but have found nothing that I could understand.  
I need to understand natural logs. Could you please explain them in 
the simplest terms possible? I don't understand what they are used 
for.  Are they used for other subjects in Math? 

Date: 12/22/97 at 02:17:21
From: Doctor Steven
Subject: Re: Natural logs

Logarithms can be pretty tricky.  The thing to remember is that a 
logarithm is just another way to write an equation that looks like 
this:

   x = a^y.

You see, normally we like to write functions like y = something, but 
here we can't do that since the y is in the exponent. So what we do is 
create some notation that will let us write this like y = something.  
This notation is the logarithm. 

We say log_a(b) is the log base a of b.
                             
And so we have y = log_a(x), which means the same thing as x = a^y .

Let's do some examples:
          
 1. x = 10^4 is the same thing as log_10(x) = 4.
 
 2. x = 10^y is the same thing as log_10(x) = y

 3. x = e^y is the same as log_e(x) = ln(x) = y.

In the third example we see that the natural log is just log base e.

Now let's get into some properties of logarithms.  Say we have 
log_a(x) = 3. This is the same as a^3 = x. Say also we have 
log_a(y) = 4 - this means a^4 = y. What is log_a(x*y)?  Well multiply 
x and y and we get x*y = a^3*a^4.

When multiplying powers we add the exponent so x*y = a^(3 + 4).  So
log_a(x*y) = 3 + 4 = log_a(x) + log_a(y).

So we have our first property of logs:

    log_a(x*y) = log_a(x) + log_a(y).

This property works for any numbers so

    log_4(20) = log_4(4*5) = log_4(4) + log_4(5) = 1 + log_4(5).

The second property is closely related to the first.  It states that:

    log_a(x/y) = log_a(x) - log_a(y).

The third property is also related to first property.  It states that:

    log_a(x^n) = n*log_a(x).

We can see this property by seeing that x^n = x*x*x*x.... (n times).
So we get log_a(x^n) = log_a(x) + log_a(x) + . . . (n times) = n*
log_a(x).

So we have 3 properties for logs:

 1. log_a(x*y) = log_a(x) + log_a(y)
 2. log_a(x/y) = log_a(x) - log_a(y)
 3. log_a(x^n) = n*log_a(x).

We use these properties to change an equation into the form we wish, 
to make it easier to work with.  

Another thing to worry about with logs is the change of base formula.  
The reason we have a change of base formula is because your calculator 
probably only has buttons for log base 10 or the natural log of a 
number.  Unfortunately not many problems will have logs that are in 
base 10 or base e, so in order to find out the exact value for these 
problems we need to change the base of the logarithm to either 10 or e 
so we can plug them into our calculator.  The change of base formula 
goes like this:

                  log_10(b)
      log_a(b) = -----------
                  log_10(a)

or
                  ln(b)
      log_a(b) = -------
                  ln(a).

An easy way to remember which number goes on top is to note the b is 
above the a on the left side of the equation and on the right side it 
is still above the a.

Let's do some examples of the change of base formula:

                 log_10(100)
 1. log_7(100) = -----------
                  log_10(7)


                  log_10(34)
 2. log_5.6(34) = -----------
                  log_10(5.6)


                   ln(1.7)
 3. log_2.3(1.7) = -------
                   ln(2.3)


You can plug these into your calculator to find the actual decimal 
value for these logarithms.

Hope this helps.                         

-Doctor Steven,  The Math Forum 
 Check out our web site!  http://mathforum.org/dr.math/