The sound intensity level was \({\beta _1}\) , and it got increased, and the value can be written as,\({\beta _2}\).
The formula for the initial level is,
\({\beta _1} = 10\log \left( {\frac{{{I_1}}}{{{I_0}}}} \right)\)
Here \({I_0}\) is the original intensity, \({I_1}\)is the measured intensity.
The expression for final intensity level is,
\({\beta _2} = 10\log \left( {\frac{{{I_2}}}{{{I_0}}}} \right)\)
Here \({I_2}\)is the measured intensity.
The increment or change is 13 dB; we can write it as,
\(\Delta \beta = {\beta _2} - {\beta _1}\)
Substitute the values in the above expression, and we get,
\(\begin{array}{c}13 = 10\log \left( {\frac{{{I_2}}}{{{I_0}}}} \right) - 10\log \left( {\frac{{{I_1}}}{{{I_0}}}} \right)\\13 = 10\log \left( {\frac{{\frac{{{I_2}}}{{{I_0}}}}}{{\frac{{{I_1}}}{{{I_0}}}}}} \right)\end{array}\)
Solving further as,
\(\begin{array}{c}13 = 10\log \left( {\frac{{{I_2}}}{{{I_1}}}} \right)\\1.3 = \log \left( {\frac{{{I_2}}}{{{I_1}}}} \right)\\\frac{{{I_2}}}{{{I_1}}} = 20\end{array}\)
Thus, the factor for the sound intensity to be increased to raise the sound intensity level by 13.0 dB is 20.