$\frac{\log \sqrt{8}}{\log 8}$ is equal to

Concept & Formula This problem tests your knowledge of the power rule of logarithms, which allows you to move an exponent from the argument to the front of the logarithm as a coefficient. $$\log (a^n) = n \log a$$ You also need to convert a square root into a fractional exponent: $\sqrt{x} = x^{\frac{1}{2}}$. Step-by-Step Solution * **Given:** $\frac{\log \sqrt{8}}{\log 8}$ * **Convert radical to exponent:** Express $\sqrt{8}$ in the numerator as a fractional power. $$\sqrt{8} = 8^{\frac{1}{2}}$$ * **Substitute this back into the expression:** $$\frac{\log (8^{\frac{1}{2}})}{\log 8}$$ * **Apply the power rule:** Move the exponent $\frac{1}{2}$ to the front of the logarithm in the numerator. $$\frac{\frac{1}{2} \log 8}{\log 8}$$ * **Simplify the fraction:** The term $\log 8$ appears in both the numerator and denominator, so they cancel out perfectly. $$\frac{1}{2}$$ Exam Strategy & Shortcut Whenever you see a logarithm of a root divided by a logarithm of the same base number, the answer is simply the fractional exponent of the root. $\sqrt{x}$ is the $\frac{1}{2}$ power, so the ratio is immediately $\frac{1}{2}$. You can bypass all intermediate written steps. Common Pitfall A major error is incorrectly simplifying the fraction by "canceling" the log terms improperly, or mistakenly trying to divide the arguments like $\frac{\sqrt{8}}{8}$. Logarithms do not work like linear algebraic variables; you must apply log properties before simplifying. Final Answer **Therefore, the correct answer is $\frac{1}{2}$.**