If ${\log }_5\left(125\right)=c$, what is the value of $c$ based on the definition of logarithm?

Simplify ${\log }_5\sqrt{x}$ using properties of logarithms.

A sound engineer combines two sound intensity ratios using decibels, which are proportional to logarithms: $0.5{\log }_{10}\left(I_1\right)+{\log }_{10}\left(I_2\right)-{\log }_{10}\left(I_3\right)$. Condense this expression into a single logarithm.

Solve for $x$: ${\log }_2\left(x^2\right)-{\log }_2(x-1)=3$

Simplify fully to a single logarithm: ${\log }_2\left(8\sqrt{x^3}\right)-\frac{1}{2}{\log }_2\left(x\right)$, assuming $x>0$.