\(\displaystyle\beta={\left({10}{d}{B}\right)}{{\log}_{{{10}}}{\left({\frac{{{I}}}{{{I}_{{{0}}}}}}\right)}}\)

If I is doubled, use a property of logarithms to get the new value of the intensity level, \(\displaystyle\beta'\). We have:

\(\displaystyle\beta'={\left({10}{d}{B}\right)}{{\log}_{{{10}}}{\left({\frac{{{2}{I}}}{{{I}_{{{0}}}}}}\right)}}{\left({10}{d}{B}\right)}{\left[{{\log}_{{{10}}}{\left({2}\right)}}+{{\log}_{{{10}}}{\left({\frac{{{I}}}{{{I}_{{{0}}}}}}\right)}}\right]}\)

\(\displaystyle={\left({10}{d}{B}\right)}{{\log}_{{{10}}}{2}}+\beta={3}{d}{B}+{65}{d}{B}={68}{d}{B}\)