why is the natural logarithm (log base) more commonly used than the logarithm base 10 in many areas of science and engineering?
In: Mathematics
10 is an artifical number, the only reason we really use it is we have 10 fingers.
e is the “natural” choice because e^x is its own derivative. The only other natural choice is 2 for computing and information theory.
There’s a reason it’s called the “natural” logarithm, and that’s because it’s base e and logarithms are related to exponents (the inverse). e is important and “natural” in math because e^x is it’s own derivative, and that’s an EXTREMELY useful fact in much of math. And because logarithms are the inverse to exponents, and e^x is such an important exponent, ln is such an important logarithm.
Note, there’s nothing magical about it, some number HAD to be, it is guaranteed, and e is that number.
Also note that log 10 is still widely used outside of mathematical equations. If you ever see a log-scaled graph and it has units, that’s going to be a log 10, not ln. As important as e is to math, it is extremely inconvenient to present your numerical results in base e rather than base 10, lol.
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.
Something also not mentioned here is that logarithm base choice hardly ever matters. We have the change of base formula to move between them if it’s ever necessary to use one in particular.
A really common phenomenon in many branches of science is that a quantity of something may change over time depending on how much of that thing there currently is. For example, populations of bacteria, the spread of disease, and returns on investment portfolios can all follow this rule. (The more bacteria present in a sample, the more they multiply, so the more bacteria are present, and so on.)
In these instances, you can predict future values using the exponential growth formula: y = C * e ^ (R * T). In this formula, C and R are known constants for your problem, T is the amount of time elapsed, and ‘e’ is Euler’s number, approximately equal to 2.718.
If you want to isolate the exponent to do something with it, you use a logarithm of the same base reverse the exponentiation. For example, the equation y = e ^ T could also be written as ln(y) = T, where ln is the logarithm base ‘e’. Because ‘e’ is used in the exponential growth formula, the log base e appears frequently in these contexts.
But why the number ‘e’ specifically. What is so special about 2.718? It has to do with something called compounding.
Imagine you have found a bank that is willing to give you 100% interest on your account every year. You put in $100.00 on January 1st and then on December 31st the bank gives you your 100% interest, giving you a total of $200.00.
Now imagine that instead of paying 100% interest once at the end of the year, the bank decides to give you 50% interest twice a year instead. On May 30th, the bank gives you your 50% interest on your $100.00 giving you $150.00 in your account. Then on December 31st the bank gives you 50% interest on your $150.00 giving you $225.00 in your account.
This is called compounding. The bank is still paying the same 100% interest rate, but the money the bank paid you in May is now also generating interest for you in December, giving you more at the end of the year. But we don’t have to stop there, if the bank compounds more frequently, you will make more money, up to a point:
1 payment of 100%: $200.00
2 payments of 50%: $225.00
4 payments of 25%: $244.14
8 payments of 12.5%: $256.57
16 payments of 6.25%: $263.79
32 payments of 3.125%: $267.69
…
1,000,000 payments of 0.0001%: $271.82
It turns out, that if the bank compounds your money as frequently as possible, at the end of the year, you will wind up with $271.82 in your account, or ‘e’ times the amount that you put in.
In the natural world, this is exactly what happens. Bacteria don’t wait a day and then double all at once, they are compounding continuously. Hence, ‘e’ becomes the natural choice for the exponential growth formula.
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