I was just musing about "log amps", where they exploit the exponential/logarithmic relationship between the current and voltage of a PN junction.
My question is, is that always "log e", ie., "ln"? My point being, is it _always_ base "e", rather than, say, log 10, or log 2, or log something else?
Is that true for silicon, germanium, gallium arsenide, every semiconductor?
Is that why they're called "natural logs"? If that's true, I actually find it kind of spooky! :-)
Thanks! Rich
It doesn't make any difference. Log, Ln they are both the same curve and vary only in constants. Remember you can change the base of a logarithm by simply dividing by the log of the other base, a constant. For example: logA = lnA/ln10. And, lnB = logB/loge. You can change to any arbitrary base, 2, hex, octal, whatever, the curve and the math is the same. The word "natural" has to do with the base e, 2.718....and its relationship to trig functions, vectors and other relations of higher mathematics, not to PN junctions. Bob
It's all the same, Rich.
log[to base B] of A = ln A / ln B
For example, if B=10, then you have:
log10 A = ln A / ln 10
But since ln 10 is a constant, it's just the same as:
log10 A = k * ln A, where k = 1.0 / ln 10
In other words, only different by a constant factor. Other than that, it's the same thing. So you can think of it this way: So long as the relationship between current and voltage in a PN junction maintains the same fixed logarithmic relationship, it's all the same. Just a change in the constant factor and that can be adduced in calibration.
It's not spooky.
And the name comes first from Nicolaus Mercator, I think. The value of 'e' itself was first found, I think, when looking at compound interest (or when my son asked me the same question he imagined by himself some years ago when he was learning about limits) -- that is, in the case of thinking about the limiting case of (1+1/x)^x as x goes to infinity.
'e^x' is:
1 + x + x^2/2! + x^3/3! + ...Taking the derivative with respect to x, you get the same series back again. If you substitute 1.0 for x, you get the value of 'e'. It also turns out that the series for sine, cosine, hyperbolic sine, and hyperbolic cosine relate very closely to this series -- especially so when you include complex numbers and the allow the imaginary value of i in x (usually using 'z' instead of 'x' for that purpose.) It plays importantly as an integrating factor for solving linear ordinary differentials, too.
Jon
log something.
natural log is base e.
Bye. Jasen
...
[excellent explanation snipped]Wow! :-)
Thanks! Rich
Interesting. I looked up compound interest and the number 2.71828 and found the formula for continuously compounding interest every nanosecond or faster.
The formula is just e^(ry) times the principal, where r is the rate and y is number of years. So, if you invest $100 at 6% for 1 year you get
100* e^(.06*1)= $106.184 which is only 18 cents more than 106 using simple interest. Not much difference. But as the time increases, the difference gets greater. $100 at 6% for 100 years compounded anually is $33930. But compounded every nanosecond, using the formula P* e^(rt) the return is $40343 for a net gain of $6413. Problem is, you have to live 100 years to collect the extra 6.4K.-Bill
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