why is the natural logarithm (log base) more commonly used than the logarithm base 10 in many areas of science and engineering?
In: Mathematics
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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