To simplify and express the given expression [tex]\(\frac{1}{2} \log_b x + 4 \log_b y - 2 \log_b x\)[/tex] as a single logarithm, follow these steps:
1. Combine the terms with [tex]\(\log_b x\)[/tex]:
[tex]\[
\frac{1}{2} \log_b x - 2 \log_b x
\][/tex]
2. Simplify the coefficients:
- Multiply each logarithm term by its coefficient:
[tex]\[
\left(\frac{1}{2} - 2\right) \log_b x = \left(\frac{1}{2} - \frac{4}{2}\right) \log_b x = -\frac{3}{2} \log_b x
\][/tex]
So, the expression becomes:
[tex]\[
-\frac{3}{2} \log_b x + 4 \log_b y
\][/tex]
3. Use the properties of logarithms to turn coefficients into exponents:
- Recall that [tex]\(a \log_b c = \log_b (c^a)\)[/tex]:
[tex]\[
-\frac{3}{2} \log_b x = \log_b (x^{-\frac{3}{2}})
\][/tex]
[tex]\[
4 \log_b y = \log_b (y^4)
\][/tex]
So, the expression now is:
[tex]\[
\log_b (x^{-\frac{3}{2}}) + \log_b (y^4)
\][/tex]
4. Combine the logarithms using the property [tex]\(\log_b a + \log_b b = \log_b (a \cdot b)\)[/tex]:
- Using the property:
[tex]\[
\log_b (x^{-\frac{3}{2}}) + \log_b (y^4) = \log_b \left(x^{-\frac{3}{2}} \cdot y^4\right)
\][/tex]
5. Express the result in simplest form:
[tex]\[
\log_b \left(\frac{y^4}{x^{\frac{3}{2}}}\right)
\][/tex]
So, the given expression [tex]\(\frac{1}{2} \log_b x + 4 \log_b y - 2 \log_b x\)[/tex] simplified and expressed as a single logarithm is:
[tex]\[
\boxed{\log_b \left(\frac{y^4}{x^{\frac{3}{2}}}\right)}
\][/tex]
