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