What are natural logarithms used for?

May 15, 2015

Hello,

1) $\ln$ is a function whose the derivative is $x \mapsto \frac{1}{x}$ on ]0,+oo[.

2) $\ln$ has the property very useful : $\setminus \ln \left({x}^{y}\right) = y \setminus \ln \left(x\right)$. Now, you can solve the equation ${3}^{x} = 2$ by using $\ln$ :
${3}^{x} = 2 \iff \ln \left({3}^{x}\right) = \ln \left(2\right) \iff x \ln \left(3\right) = \ln \left(2\right) \iff x = \ln \frac{2}{\ln} \left(3\right)$.

3) In Chemical, you know the formula $\left[{H}_{3} {O}^{+}\right] = {10}^{- p H}$. Therefore, $\ln \left(\left[{H}_{3} {O}^{+}\right]\right) = - p H \ln \left(10\right)$, and
$p H = - \ln \frac{\left[{H}_{3} {O}^{+}\right]}{\ln} \left(10\right)$.
Remark. $\ln \frac{x}{\ln} \left(10\right)$ is denoted $\log \left(x\right)$.

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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