What are natural logarithms used for?

1 Answer
vince
May 15, 2015

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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