(d) \(\frac{3}{2}\)
\(2^{log_{10}3\sqrt3}\) = \(3^{k\,log_{10}}\)2 ⇒ \(2^{log_{10}\big(3^{\frac{3}{2}}\big)}\) = \(3^{k\,log_{10}}\)2
⇒\(2^{log_{2}\big(3^{\frac{3}{2}}\big).log_{10^2}}\) = \(3^{k\,log_{10}}\)2 [Using logax = logbx .logab]
⇒ \(\bigg[2^{log_{2}\big(3^{\frac{3}{2}}\big)}\bigg]^{log_{10^2}}\) = \((3^{k})^{log_{10}}\)2 ⇒ \(2^{log_2\,3^{\frac{3}{2}}}\) = 3k
⇒\(3^{\frac{3}{2}}\) = 3k ⇒ k = \(\frac{3}{2}\) (∵ \(a^{log_a\,x}\) = x)