RM{a}}}{\rm{B}} = \frac{{{{\log }_{\rm{e}}}{\rm{b}}}}{{{{\log }_{\rm{e}}}{\rm{a}}}}\)\({\log _{\rm{a}}}{\rm{b}} = \frac{1}{{{{\log }_{\rm{b}}}{\rm{a}}}}\)Where a ≠ 1, a > 0 and b ≠ 1, b > 0 and M, N are arbitrary positive numbers and p is any real number.Calculation:Given: f(x) = log10 (5X2 + 3)To find: f’(x)We know that, \({\log _{\rm{a}}}{\rm{b}} = \frac{{{{\log }_{\rm{e}}}{\rm{b}}}}{{{{\log }_{\rm{e}}}{\rm{a}}}}\)So, \({\rm{f}}\left( {\rm{x}} \right) = {\rm{\;}}\frac{{{{\log }_{\rm{e}}}\left( {5{{\rm{x}}^2} + 3} \right)}}{{{{\log }_{\rm{e}}}10}}\)Differentiating both sides w.r.t x\(\Rightarrow {\rm{f'}}\left( {\rm{x}} \right) = {\rm{\;}}\frac{1}{{\left( {5{{\rm{x}}^2}\; + \;3} \right)\; \times \;{{\log }_{\rm{e}}}10}} \times 10{\rm{x}}\)Using property, \({\log _{\rm{a}}}{\rm{b}} = \frac{1}{{{{\log }_{\rm{b}}}{\rm{a}}}}\)\( \Rightarrow {\rm{f'}}\left( {\rm{x}} \right) = \frac{{10{\rm{x}}{{\log }_{10}}{\rm{e}}}}{{5{{\rm{x}}^2}\; + \;3}}\)