Question

Solve for $x$, if
$x+{\log }_{10}(1+2^x)=x{\log }_{10}5+{\log }_{10}6$

Answer 1

$x+lo{g}_{10}(1+2^x)=xlo{g}_{10}5+lo{g}_{10}6$

$x+lo{g}_{10}(1+2^x)=lo{g}_{10}\left({6.5}^x\right)$

$x=lo{g}_{10}(6\times 5^x)-log(1+2^x)$

$x=lo{g}_{10}\left(\frac{6\times 5^x}{1+2^x}\right)$

${10}^x=\frac{6\times 5^x}{1+2^x}$

$\Rightarrow 2^x\times 5^x=\frac{6\times 5^x}{1+2^x}$

Put $2^x=t$

$t(1+t)=6$

$\Rightarrow t(1+t)=2\times 3$

$\Rightarrow t=2$

$\Rightarrow 2^x=2$

$\Rightarrow x=1$

$\therefore x=1$.