What are natural logarithms used for?

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What are natural logarithms used for?

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Answer 1 · 2015-05-15T16:08:12

Hello,

There are lots of answers.

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

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

3) In Chemical, you know the formula $[H_{3} O^{+}] = 10^{- p H}$ $[H_3O^+] = 10^{-pH}$ . Therefore, $ln ([H_{3} O^{+}]) = - p H ln (10)$ $ln([H_3O^+]) = -pH ln(10)$ , and
$p H = - \frac{ln ([H_{3} O^{+}])}{ln (10)}$ $pH = - ln([H_3O^+])/ln(10)$ .
Remark. $\frac{ln (x)}{ln (10)}$ $ln(x)/ln(10)$ is denoted $log (x)$ $log(x)$ .

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

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

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What are natural logarithms used for?

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