To express [tex]\(\frac{1}{2} \log a + 3 \log b \)[/tex] as a single logarithm and simplify it, follow these steps:
1. Identify the given logarithmic expressions:
[tex]\[
\frac{1}{2} \log(a) \quad \text{and} \quad 3 \log(b).
\][/tex]
2. Apply the power rule of logarithms, which states that [tex]\( k \log(x) = \log(x^k) \)[/tex].
For [tex]\(\frac{1}{2} \log(a)\)[/tex], we have:
[tex]\[
\frac{1}{2} \log(a) = \log(a^{\frac{1}{2}}).
\][/tex]
For [tex]\(3 \log(b)\)[/tex], we have:
[tex]\[
3 \log(b) = \log(b^3).
\][/tex]
3. Combine the logarithmic expressions using the logarithm addition rule, which states that [tex]\( \log(x) + \log(y) = \log(xy) \)[/tex].
Thus, we combine [tex]\(\log(a^{\frac{1}{2}})\)[/tex] and [tex]\(\log(b^3)\)[/tex]:
[tex]\[
\log(a^{\frac{1}{2}}) + \log(b^3) = \log(a^{\frac{1}{2}} \cdot b^3).
\][/tex]
4. Simplify the expression if necessary. In this case, the expression is already simplified.
Therefore, the expression [tex]\(\frac{1}{2} \log a + 3 \log b\)[/tex] as a single logarithm is:
[tex]\[
\boxed{\log(a^{\frac{1}{2}} \cdot b^3)}.
\][/tex]
