Evaluate log base 1/4 of 2

Rewrite as an equation.

Rewrite ${\displaystyle {\log }_{\frac{1}{4}}\left(2\right)=x}$ in exponential form using the definition of a logarithm. If ${\displaystyle x}$ and ${\displaystyle b}$ are positive real numbers and ${\displaystyle b}$ does not equal ${\displaystyle 1}$, then ${\displaystyle {\log }_b\left(x\right)=y}$ is equivalent to ${\displaystyle b^y=x}$.

Create expressions in the equation that all have equal bases.

Rewrite ${\displaystyle {\left(2^{-2}\right)}^x}$ as ${\displaystyle 2^{-2x}}$.

Since the bases are the same, then two expressions are only equal if the exponents are also equal.

Solve for ${\displaystyle x}$.

The variable ${\displaystyle x}$ is equal to ${\displaystyle -\frac{1}{2}}$.