To simplify the logarithmic expression [tex]\(\log _b \sqrt[6]{\frac{x^5}{y^6 z^7}}\)[/tex] and express it as a sum or difference of logarithms without exponents, let's follow these steps carefully:
1. Start with the given expression:
[tex]\[
\log_b \sqrt[6]{\frac{x^5}{y^6 z^7}}
\][/tex]
2. Recognize that [tex]\(\sqrt[6]{a}\)[/tex] is equivalent to [tex]\(a^{1/6}\)[/tex]. Apply this to the expression:
[tex]\[
\log_b \left( \frac{x^5}{y^6 z^7} \right)^{1/6}
\][/tex]
3. Use the exponent rule of logarithms: [tex]\(\log_b (a^c) = c \cdot \log_b (a)\)[/tex]. Here, [tex]\(c = \frac{1}{6}\)[/tex]:
[tex]\[
\frac{1}{6} \cdot \log_b \left( \frac{x^5}{y^6 z^7} \right)
\][/tex]
4. Apply the quotient rule of logarithms: [tex]\(\log_b \left( \frac{a}{b} \right) = \log_b (a) - \log_b (b)\)[/tex]:
[tex]\[
\frac{1}{6} \cdot (\log_b (x^5) - \log_b (y^6 z^7))
\][/tex]
5. Apply the product rule of logarithms inside the parentheses: [tex]\(\log_b (a \cdot b) = \log_b (a) + \log_b (b)\)[/tex]:
[tex]\[
\frac{1}{6} \cdot \left( \log_b (x^5) - (\log_b (y^6) + \log_b (z^7)) \right)
\][/tex]
6. Apply the exponent rule again to each term: [tex]\(\log_b (a^c) = c \cdot \log_b (a)\)[/tex]:
[tex]\[
\frac{1}{6} \cdot \left( 5 \cdot \log_b (x) - (6 \cdot \log_b (y) + 7 \cdot \log_b (z)) \right)
\][/tex]
7. Distribute the [tex]\(\frac{1}{6}\)[/tex] across the terms inside the parentheses:
[tex]\[
\frac{1}{6} \cdot 5 \cdot \log_b (x) - \frac{1}{6} \cdot 6 \cdot \log_b (y) - \frac{1}{6} \cdot 7 \cdot \log_b (z)
\][/tex]
8. Simplify the coefficients:
[tex]\[
\frac{5}{6} \cdot \log_b (x) - \log_b (y) - \frac{7}{6} \cdot \log_b (z)
\][/tex]
So, the equivalent sum or difference of logarithms without exponents is:
[tex]\[
\boxed{\frac{5}{6} \log_b (x) - \log_b (y) - \frac{7}{6} \log_b (z)}
\][/tex]
