Evaluate the given logarithm.
${\log }_y\sqrt{y}$

• A. $2$
• B. $\frac{1}{2}$
• C. $y^2$
• D. $y^{\frac{1}{2}}$

Answer: (B)

Verified step by step guidance

1. Recall the logarithm property that relates the logarithm of a power: \(\log_{a}(b^{c}) = c \cdot \log_{a}(b)\).
2. Rewrite the expression \(\log_{y}(\sqrt{y})\) by expressing the square root as an exponent: \(\sqrt{y} = y^{\frac{1}{2}}\).
3. Apply the logarithm power rule to get \(\log_{y}(y^{\frac{1}{2}}) = \frac{1}{2} \cdot \log_{y}(y)\).
4. Use the fact that \(\log_{a}(a) = 1\) for any positive base \(a \neq 1\), so \(\log_{y}(y) = 1\).
5. Multiply the coefficient by 1 to simplify the expression to \(\frac{1}{2} \times 1 = \frac{1}{2}\), which is the value of the logarithm.