Question

What are natural logarithms used for?

Answer 1

Hello,

There are lots of answers.

1) `ln` is a function whose the derivative is `x mapsto 1/x` on `]0,+oo[`.

2) `ln` has the property very useful : `\ln(x^y) = y\ln(x)`. Now, you can solve the equation `3^x = 2` by using `ln` :
`3^x = 2 <=> ln(3^x) = ln(2) <=> x ln(3) = ln(2) <=> x = ln(2)/ln(3)`.

3) In Chemical, you know the formula `[H_3O^+] = 10^{-pH}`. Therefore, `ln([H_3O^+]) = -pH ln(10)`, and
`pH = - ln([H_3O^+])/ln(10)`.
Remark. `ln(x)/ln(10)` is denoted `log(x)`.

4) `ln` is used for the Richter magnitude scale to quantify the earthquake.

5) `ln` is used in decibel scale to quantify the noise.

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