Common Logarithims and Natural Logarithims

Date: 03/13/2005 at 13:18:50
From: Katie
Subject: Common logarithims vs. Natural logarithims

I am currently studying logarithims and I saw that logarithims can 
take the form of ln or log.  What is the difference between the two?  

I think it's very confusing because I looked it up in my math book and 
they state log e x = ln x.  Then they state log 10 x =log x.  I am 
confused about how e and 10 work with ln and log.  

I think that if I see a problem such as y= ln (x-1) and the book asks 
to find the inverse, I have to change it to log, but I don't know how 
to do that.  I simply don't understand ln and log's relationship.

Date: 03/13/2005 at 14:46:00
From: Doctor Tom
Subject: Re: Common logarithims vs. Natural logarithims

Hi Katie,

You're confused because it really is confusing!

When logarithms are first introduced, it is much easier for students
to think about logarithms in base 10.  In base 10,

  log(.01) = -2
  log(.1) = -1
  log(1) = 0
  log(10) = 1
  log(100) = 2
  log(1000) = 3

an so on--it just sort of counts the zeros, or, more accurately, the
log is the power of 10 that creates the number you are taking the log
of.  Since 100 is 10^2, the log of 100 is 2.

Engineers also tend to use log base 10 for most calculations for the
same reason.  You can just look at the size and know the magnitude of 
the number.  if the log (base 10) is 5.46, without even thinking, you 
know the number has "between" 10^5 and 10^6, so it's between 100,000 
and 1,000,000.

So in introductory texts and in engineering books, when you see "log",
it usually means "log base 10".  If they DO want to talk about other
bases, they either put a little subscript after the "log", like
"log_e".  (I can't draw a subscript, so I'm using the "_" to mean that 
the next character should be smaller and written as a subscript).

Since log base e is often important for engineers, particularly 
electrical engineers, they often use "ln" instead of "log_e" since 
it's quicker to write, and it's a mnemonic for "logarithm, natural",
or "natural logarithm".

Now for mathematicians, the "natural log" really IS much more natural,
so since that's the ONLY type of logarithm they use, they often just
write "log" instead of "ln".  I know this seems really confusing, but
if you know what you're doing, you can almost always tell that it's a
natural log just by looking at the equation.  Of course if there is a
chance of confusion, everybody always writes them like "log_10" or
"log_e" to make it obvious what's going on.

Since you are just starting to learn about logarithms, you will always
see "log" as meaning "log_10".

Computer scientists often have a very good use for "log base 2", or
"log_2", and it comes up so often there that they often write "lg"
instead of "log_2".  I've never seen them shorten "lg" or "log_2" to
just "log" like the mathematicians, however, so you're never in any
danger there.

It turns out that if you want to switch bases, from one base to 
another, you just multiply by a constant that depends on the bases
involved.

For example:

  log_10(x) = .4342944819... log_e(x)

  The value .4342944819... is just log_10(e)

Here's the general rule:

  log_a(b) = log_a(c) * log_c(b)

You can check this just from the definitions of logarithms.

If you ever get confused about logarithms, like you might above, you
can just convert to the exponential form.

To prove the relationship above, for example, just start this way:

  x = log_a(b)  means a^x = b
  y = log_a(c) means  a^y = c
  z = log_c(b) means  c^z = b

Since c^z = b and  c = a^y, we have (a^y)^z = b or a^(yz) = b

But a^x = b also, so x = yz.

And that's just what we were trying to show.

- Doctor Tom, The Math Forum
  http://mathforum.org/dr.math/