Question

If $a^{x} = b^{y} = (ab)^{xy},$ then $x + y =$

• A. 0
• B. 1
• C. xy
• D. None of these

Answer 1

Answer: (B)

1

Answer 2

$a^{x} = b^{y} = (ab)^{xy}$

$\Rightarrow x\ln a = y\ln b = xy\ln(ab) = k(\text{say})$

$\ln a = \frac{k}{x},\ln b = \frac{k}{y}$

$\ln(ab) = \frac{k}{xy} \Rightarrow$ $\ln a + \ln b = \frac{k}{xy}$ $\Rightarrow$ $\frac{k}{x} + \frac{k}{y} = \frac{k}{xy}$

$\Rightarrow$ $\frac{1}{x} + \frac{1}{y} = \frac{1}{xy}$ $\Rightarrow$ $\frac{x + y}{xy} = \frac{1}{xy}$; $\therefore$ $x + y = 1$.