I don’t know what these functions actually mean or what they stand for. Someone fill me in.
e is a number (a transcendental number with a value of ~2.7), it is primarily of interest because of the exponential function which turns up alot in maths.
f(x) = exp(x) = e[sup]x[/sup] = f’(x).
The general solution to the most basic differential equation:
f(x) - f’(x) = 0
is Ae[sup]x[/sup]
Where A is any constant.
It’s also related to the trigonmetric functions by:
e[sup]ix[/sup] = cos x + isin x
ln(x) inverse of the functiuon f(x) = e[sup]x[/sup] i.e.
e[sup]ln(x)[/sup] = x
and also has many applications. ln(x) is called the natural logaritm, or the logaritm to the base e of x
The log(x) to the base a is the inverse of the function f(x) = a[sup]x[/sup] therefore:
a[sup]log(x)[/sup] = x.
(when the ‘L’ is captialed as you have done in your title (i.e. Log(x)) it usually means the primary branch of the function f(x) = log(x))
Here’s a decent website on the subject:
e is the number that is computed by the following expression when you take the limit as x increases without limit (i.e. gets bigger and bigger):
(1+1/x)[sup]x[/sup]
when x = 10 the result is 2.5937424601
when x = 100 … 2.70481382942153
when x = 1000 … 2.7692393223559
when x = 10000 … 2.71814592682493
when x = 100000 … 2.7182682371923
when x = 1000000 … 2.71828046909575
And and x gets really really big it comes out to 2.71818182845905…
As was stated in another post, this ratio and this number arises in a lot of applications of more advanced mathematical studies and mathematical applications.
Logarithms began as an aid to computation. A log is the exponent of a number, called the base of the logaritm. They were handy for computation in the days before computers because they reduce multiplication to addition, division to subtraction and raising to a power to multiplication.
The two most used bases are e and 10. In US usage logs to base e are natural logs and logs to base 10 are common logs.
How do they work?
If I multiply x[sup]1[/sup] by x[sup]2[/sup] I add the exponents and get x[sup]3[/sup]. Now suppose x is 10. 10[sup]1[/sup]* 10[sup]1[/sup] = 10[sup]1[/sup] or 10*100 = 1000.
I can now form a table:
Number: Logarithm:
10 ------------------- 1
100 ------------------ 2
1000 ----------------- 3
Now if I can’t multiply but can add I can add and have the table I look up the log of 10 and 100 which are 1 and 2, add them to get three which is the logarithm of my answer. So I find 3 in the logarithm column and sure enough the answer is 1000.
For division I subtract because x[sup]3[/sup]/ x[sup]2[/sup] = x[sup]3-2[/sup] = x[sup]1[/sup]. Using my table again I find the log of 1000 which is 3, the log of 100 which is 2, subtract them and I have 1 which is the log of the answer. 1000/100 = 10.
Taking a root is now done by dividing the log by the root. For example the square root of 100 is 10. The log of 100 is 2 divided by the root 2 gives 1 which is the log of 10 which was the answer.
That’s fine for whole number logs but how do I fill in the spaces? Well the roots give us one way to do it. The square root of 1000 is 31.623 to three places and its log is the log of 1000 or 3 divided by 2 so the log of 31.623 is 1.5 and I have another number pair for the table. I can find the square root of 1000 by the drudgery of trial and error. I just keep trying numbers to more and more decimal places multiplied by themselves until I get 1000 to the degree of accuracy I need for my purposes.
I think you can now see that I can take the cube root of 100 which is 4.642 and its log is the log of 100, 2, divided by 3 or 0.66667. And so on.
There are much more elegant ways to find the logs of number than using trial and error but all of them involved laborious hand computation and the early workers with logarithms worked out logs to 6 or 7 decimal places by hand computations. However, once that was done, computation of complex problems was made a lot easier.
Oh my! In this line: Now suppose x is 10. 10[sup]1[/sup]* 10[sup]1[/sup] = 10[sup]1[/sup] or 10*100 = 1000.
Make it read: Now suppose x is 10. 10[sup]1[/sup] 10[sup]2[/sup] = 10[sup]3[/sup]* or 10*100 = 1000.
For what it’s worth, here are the traditional definitions.
exp(x) = sum(x[sup]n[/sup]/n!, 0 < n).
ln(x) = integral(1/t, 1 < t < x).
a[sup]b[/sup] is defined as exp(b*ln(a)).
Alrighty, that made no sense. I did just learn that y=ln(x) is the same as e^y=x. And x=loga^b is the same as a^x=b. So if I need to solve for the exponent they come in handy.
Before e and ln make any sense, you need to understand log.
From my maths teacher:
Repeat incessently, until it somehow begins to make sense. That takes some time. It’s a strange concept, and there’s no non-mathematical visual handy mnemonical metaphor that will help explain it. Keep thinking that sentence over until it makes sense. Then tackle ln and e. 
“That made no sense” isn’t much of an indication of what it is that made no sense.
And would you expand a little on what you mean by “… if I need to solve for the exponent they come in handy.”? What would you do to actually solve for the exponent in the case of 41 = 10[sup]x[/sup] or 2 = e[sup]x[/sup]?
In the use of logarthms you do not solve for the exponent. The exponent is the logarithm and you use that to make other computations such as multiplication, division, raising to a power or extracting roots.
I know this sounds cumbersome to look up two logs, add or subtract them and then look up the answer when you can just punch a few buttons on a calculator. Before calculators computation wasn’t that simple. And for those who aren’t mathematicians and have no calculator being able to extract a cube root by simply dividing its logarithm by 3 and getting the result out a table makes life easy.
Wesley Clark, I don’t know how much interest this will be to you, but I thought it might be interesting for you to see a couple of ways the number e pops up in applications (and why this number is useful).
Are you familiar with compound interest?
If you invest $1.00 at 100% interest compounded daily (365 times a year), at the end of the year your investment is worth $2.7145674820218743031938863066851… (of course, in the real world this will be rounded off to an exact cent).
On the other hand, if it’s compounded hourly (8760 times a year), at the end of the year your investment is worth 2.718126691620452118916138065396…
Compounded every second (31,536,000 times a year), by the end of the year your investment is worth $2.7182817853609708212635582662979…
It looks like those numbers are getting closer and closer to a particular number. We define “compounded continuously” so that the value of the investment is that particular number which the above numbers are converging to. In other words, $1 invested at 100% interest compounded continuously is $2.7182818284590452353602874713526… This is exactly e dollars, one way in which this number pops up in a natural setting.
One more example. Say you’re at a bar with about a dozen friends. The waitress comes to your table and takes everybody’s drink order (let’s say everybody orders a different drink). When she returns, her memory is completely shot, she has no idea who ordered what. Instead of asking each person what he/she ordered, she just randomly sets a drink down in front of each person, and walks away. What’s the probability that nobody gets the correct drink? About 1/e = 36.787944117144232159552377016146…% (the more friends you’re with, the closer the actual probability will be to 1/e).
I think you’re both right. If you took the equation 2 = e[sup]x[/sup] and solved it for the exponent x, you’d have x = ln 2.
Or, say something increases by 5% each year; how long will it take to double? To answer this you’d have to solve 2 = 1.05[sup]x[/sup].
Then x = log[sub]1.05[/sub] 2 = (ln 2)/(ln 1.05).
Good points. It might be a good idea to go on and show how the last equality comes about.
given x = log[sub]1.05[/sub] 2 this is the same as 2 = 1.05[sup]x[/sup].
Take the natural log of both sides: ln(2) = x*ln(1.05)
and x = ln(2)/ln(1.05)
Which is a good form in which to put the answer because tables or computer programs that give log[sub]1.05[/sub] are hard to find.
David Simmons
*
Which is a good form in which to put the answer because tables or computer programs that give log1.05 are hard to find.*
On my website I happen to have written a program that will compute logs for any base.
http://www.1728.com/logrithm.htm
(Oh it finds the anti-logs too).
And Wesley Clark, here’s a compound interest calculator:
http://www.1728.com/compint.htm
Ahhhh but it doesn’t show the compounding method. Well then this calculator figures it out for semi-annually, quarterly … all the way to continuously.
http://www.1728.com/yieldint.htm
Sorry for showing off but I think it is relevant to this discussion.
‘e’
<http://www.math.toronto.edu/mathnet/answers/ereal.html>
‘log’ or logarithm also ‘Ln’ or natural log
<http://www.fact-index.com/l/lo/logarithm.html>
Search engines are a wonderful thing if you take the time.
One possible confusion to be aware of if you need to program using logs: In writing, without the base specified, log(x) = log[sub]10/sub, and ln(x) = log[sub]e/sub, but in programming (at least in the languages I’m familiar with), it’s instead log(x) = log[sub]e/sub and log10(x) = log[sub]10/sub. You may also run across exp(x) which is just e[sup]x[/sup].
Whoa! Let’s back up ten paces. If Wesley Clark is asking what they mean, I assume he doesn’t really need to know (yet) the power series expansion of e.
Logarithms come from two observations. First, if 10 = 101 and 100 = 102, we can guess that a number between 10 and 100, say x, can be expressed as 10**y, where y is between 1 and 2. Okay?
Second, as I’m sure you learned sometime, ab * ac can be written as a ** (b+c). So you can multiply two numbers with the same base by adding their exponents. Since any number can be written as 10**z, you can multiply two numbers by adding their exponents. The log of a number, base 10, is just the exponent you need to raise 10 to to get the number. All the other stuff people have written follows from this.
This was particularly handy in the dark ages, when I went to school, before there were calculators. A slide rule is just a way of adding two exponents, to do multiplication quickly and easily by manipulating the slipstick.
Now what is this e business? You don’t have to use 10 as the base - you can use any number. It turns out that using e (2.7 … it’s irrational) has some interesting properties, and shows up all the time in nature, such as in the definition of sine and cosine. Thus it gets used all the time. ln, the natural logarithm, is just log base e.
Hope this helps. If anything isn’t clear, let me know.
One thing to be aware of notation is never 100% standardized so log(x) without the subscript denoting the base can mean different things to different people. To a physcist log(x) (without the base) is usually the logaritm to the base 10 of x, but to a mathematican it is often the natural logarithm of x.
Going off at a tangent, the algebraic properties of logaritms as alluded to by Voyager, namely:
log[sub]a/sub + log[sub]a/sub = log[sub]a/sub
where a, x and y are postive real numbers and a > 1.
Show that addition in the reals and multiplication in the postive reals (which both satisify the axioms of a group) are isomorphic.
That depends on the context. If it’s a label for the axis of a graph, then “log(x)” would mean base 10, but if it’s in an equation, it probably means natural log. To further confuse matters, if one is speaking an equation, even if it’s written “ln(x)”, one would say that as “log x”.
Incidentally, another base often used for logarithms is 2, especially in computer science. log[sub]2/sub tells you how many bits you need to represent x, and many basic algorithms will require log[sub]2/sub (or some multiple of that) operations.
In every lower level math textbook that I have seen (up through at least Calculus), log refers to the “common logarithm” (base 10—which was admittedly much more common before electronic calculators were readily available). But in more advanced math (e.g. discussion of the Prime Number Theorem) log is often used instead of ln to denote the natural logarithm (base e).
Every scientific calculator I’ve ever seen has both a log button (for log[sub]10[/sub]) and an ln button. But (as ZenBeam noted), log gives the natural logarithm in many computer programming contexts.
To make matters even more interesting, when dealing with complex variables, Log or Ln is capitalized to denote the principal branch of the logarithm function.
I’ve always pronounced “ln(x)” as “el en of x” or “the natural log of x.”
I’ve been in that bar! 