Question

Find the sum of ‘n’ terms of the series :

\(\text{log}_2\big(\frac{x}{y}\big)\) + \(\text{log}_4\big(\frac{x}{y}\big)^2\) + \(\text{log}_8\big(\frac{x}{y}\big)^3\) + \(\text{log}_{16}\big(\frac{x}{y}\big)^4\) + .............

(a) \(\text{log}_2\big(\frac{x}{y}\big)^{4n}\)

(b) \(n\,\text{log}_2\big(\frac{x}{y}\big)\)

(c) \(\text{log}_2\big(\frac{x^{n-1}}{y^{n-1}}\big)\)

(d) \(\frac{1}{2}\text{log}_2\big(\frac{x}{y}\big)^{n(n+1)}\)

Answer 1

(b) \(n\,\text{log}_2\big(\frac{x}{y}\big)\)

Given series

=  \(\text{log}_2\big(\frac{x}{y}\big)\) + \(\text{log}_{2^2}\big(\frac{x}{y}\big)^2\) + \(\text{log}_{2^3}\big(\frac{x}{y}\big)^3\) + \(\text{log}_{2^4}\big(\frac{x}{y}\big)^4\) + .............

= \(\text{log}_2\big(\frac{x}{y}\big)\) + \(\text{log}_2\big(\frac{x}{y}\big)\) + \(\text{log}_2\big(\frac{x}{y}\big)\) + \(\text{log}_2\big(\frac{x}{y}\big)\) + .......n terms

= \(\text{log}_2\big(\frac{x}{y}.\frac{x}{y}.\frac{x}{y}.\frac{x}{y}.\,.......n \,\text{terms}\big)\)

= \(\text{log}_2\big(\frac{x}{y}\big)^n\) = \(n\,\text{log}_2\big(\frac{x}{y}\big)\).