(10x−(10−x)11=12).

${\displaystyle (\frac{{10}^x-\left({10}^{-x}\right)}{11}=12).}$
${\displaystyle (({10}^x-\left({10}^{-x}\right))\cdot 11=12\cdot 11}$
${\displaystyle \left({10}^x\right)-\left({10}^{-x}\right)=132}$
${\displaystyle {10}^x-{\left({10}^x\right)}^{-1}=132}$
Rewrite the equation with ${\displaystyle {10}^x=u}$
${\displaystyle u-{\left(u\right)}^{-1}=132}$
${\displaystyle u-\left(\frac{1}{u}\right)=132}$
${\displaystyle u-\left(\frac{1}{u}\right)u=132u}$
${\displaystyle u^2-1=132u}$
${\displaystyle u^2-132u-1=0}$
${\displaystyle u=66+\sqrt{4357},u=66-\sqrt{4357}}$
Substitute back ${\displaystyle u={10}^x}$, solve for x
Solve ${\displaystyle {10}^x=66+\sqrt{4357};x=\frac{\ln (66+\sqrt{4357})}{\ln \left(10\right)}}$
Solve ${\displaystyle {10}^x=66-\sqrt{4357}}$; No Solution for $x\in Rx=(\ln (66+\sqrt{4357}))/\ln (10)$