If 2^(log10 cube root (3)) = 3^(klog10)^2 then the value of k is :

Question

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 1

(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)