If \(2^{log_{10}3\sqrt3}\) = \(3^{k\,log_{10}}\)2 then the value of

1.	If \(2^{log_{10}3\sqrt3}\) = \(3^{k\,log_{10}}\)2 then the value of k is :(a) 1 (b) \(\frac{1}{2}\)(c) 2 (d) \(\frac{3}{2}\)
Answer» (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 log_ax = log_bx .log_ab] ⇒ \(\bigg[2^{log_{2}\big(3^{\frac{3}{2}}\big)}\bigg]^{log_{10^2}}\) = \((3^{k})^{log_{10}}\)²⇒ \(2^{log_2\,3^{\frac{3}{2}}}\) = 3^k ⇒\(3^{\frac{3}{2}}\) = 3^k⇒ k = \(\frac{3}{2}\) (∵ \(a^{log_a\,x}\) = x)

If \(2^{log_{10}3\sqrt3}\) = \(3^{k\,log_{10}}\)2 then the value of k is :(a) 1 (b) \(\frac{1}{2}\)(c) 2 (d) \(\frac{3}{2}\)

Answer»

(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 log_ax = log_bx .log_ab]

⇒ \(\bigg[2^{log_{2}\big(3^{\frac{3}{2}}\big)}\bigg]^{log_{10^2}}\) = \((3^{k})^{log_{10}}\)²⇒ \(2^{log_2\,3^{\frac{3}{2}}}\) = 3^k

⇒\(3^{\frac{3}{2}}\) = 3^k⇒ k = \(\frac{3}{2}\) (∵ \(a^{log_a\,x}\) = x)