Answer
$x=\sqrt{\frac{10^{11}}{2}}\approx 223606.7977$
Work Step by Step
Recall the product property of logarithms (pg. 462):
$\log_b{mn}=\log_b{m}+\log_b{n}$
Applying this property to our equation, we get:
$\log (2x\cdot x)=11$
$\log{(2x^2)}=11$
Next, recall the definition of a logarithm (pg. 451):
$\log_{b}{x}=y$ iff $b^y=x$
Applying this definition to our equation (with $b=10, y=11, x=2x^2$), we get:
$10^{11}=2x^2$
$x^2=\dfrac{10^{11}}{2}$
$x=\sqrt{\frac{10^{11}}{2}}\approx 223606.7977$
We confirm that the answer works:
$\log {\left(2\cdot 223606.7977\right)}+ \log {223606.7977}=11$
$\log{447213.7977}+ \log{223606.7977}=11$
$5.65+5.35=11$
$11=11$