Question

Write explicitly, functions of $y$ defined by the following equations and also find the domains of the given implicit functions :

${10}^x+{10}^y=10$

• A. $y={\log }_{10}(10-{10}^x),-\infty <x<1$
• B. $y={\log }_{10}(10+{10}^x),-\infty <x<1$
• C. $y={\log }_{10}(10+{10}^x),-\infty <x\le 1$
• D. None of these

Answer 1

The correct option is A $y={\log }_{10}(10-{10}^x),-\infty <x<1$

${10}^x+{10}^y=10$
${10}^y=10-{10}^x$
$y={\log }_{10}(10-{10}^x)$
Domain :
For $y$ to be defined
$10-{10}^x>0$
$\Rightarrow 10>{10}^x$
$\Rightarrow x<1$
$\Rightarrow x\in (-\infty ,1)$