why is the natural logarithm (log base) more commonly used than the logarithm base 10 in many areas of science and engineering?
In: Mathematics
Its because of the exponential function. We like exp because:
1) its defining property is that exp(a+b)=exp(a)exp(b) usually very neat
2) its derivitive is itself, its a derivitive fix point.
As a consequence of that when we have differential equations that are something on the line of: the derivitive of the function is proportional to itself we get exp as a solution. And we often deal with diff equations like that.
What if we used 10 instead of e? Lets look at the derivitive of 10^(x). We can rewrite a power like a^(b) = e^b(lna) = e^(lna)^(b) = a^(b). So we can take the derivitive of 10^(x) = e^(xln(10)) = exp(x × ln(10)) and its derivitive is exp(ln10 × x) × ln(10) = 10^x × ln 10. (And its not like ln is avoidable here, the whole point is that you can redefine powers with a workable function.)
As you can see using anything other than e^(x) for an exponential is an unnecessary complication. e^(x) is just more convenient function to work with and so its inverse also shows up more frequently than others base logarithms. Thats why ln is called the natural base log or natural log, thats what the n stands for. Its in a sense more natural to use as anything other compared to it is just a handicap.
Of course in computer science base 2 logs can show up quite frequently if you insist on thinking in terms of bits.
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