Question

If $y$ is a function of $x$ defined by $a^{x+y}=a^x+a^y$ where $a$ is a real constant $(a>1)$ then the domain of $y\left(x\right)$ is

• A. $(0,+\infty )$
• B. $(-\infty ,0)$
• C. $(-1,+\infty )$
• D. $(-\infty ,1)$

Answer 1

The correct option is A $(0,+\infty )$

$a^{x+y}=a^x+a^y$

$1=a^{-x}+a^{-y}$

$a^{-y}=1-a^{-x}$

Taking ${\log }_a$ on both sides, we get

$-y={\log }_a(1-a^{-x})$

$y=-{\log }_a(1-a^{-x})$

Now

$1-a^{-x}>0$

$1>a^{-x}$

$a^x>1$

$x>0$

Hence domain for the above function is $(0,\infty )$.