To expand the expression [tex]\(\log \left(\frac{x^6 y}{z}\right)\)[/tex], you can use the properties of logarithms. Here’s the detailed, step-by-step solution:
1. Understand the given expression: We start with the logarithmic expression [tex]\(\log \left(\frac{x^6 y}{z}\right)\)[/tex].
2. Apply the quotient rule of logarithms: The quotient rule of logarithms states that [tex]\(\log \left(\frac{a}{b}\right) = \log(a) - \log(b)\)[/tex]. Applying this rule, we get:
[tex]\[
\log \left(\frac{x^6 y}{z}\right) = \log (x^6 y) - \log (z)
\][/tex]
3. Apply the product rule of logarithms: The product rule of logarithms states that [tex]\(\log (a \cdot b) = \log (a) + \log (b)\)[/tex]. Applying this rule to the term [tex]\(\log (x^6 y)\)[/tex], we get:
[tex]\[
\log (x^6 y) = \log (x^6) + \log (y)
\][/tex]
4. Simplify the logarithm of a power: The power rule of logarithms states that [tex]\(\log (a^b) = b \cdot \log (a)\)[/tex]. Applying this rule to the term [tex]\(\log (x^6)\)[/tex], we get:
[tex]\[
\log (x^6) = 6 \cdot \log (x)
\][/tex]
5. Combine all the steps together: Substitute the simplified terms back into the original expression:
[tex]\[
\log \left(\frac{x^6 y}{z}\right) = \log (x^6 y) - \log (z)
\][/tex]
[tex]\[
\log (x^6 y) = \log (x^6) + \log (y)
\][/tex]
[tex]\[
\log (x^6) = 6 \cdot \log (x)
\][/tex]
Therefore, the expanded expression is:
[tex]\[
\log \left(\frac{x^6 y}{z}\right) = 6 \cdot \log (x) + \log (y) - \log (z)
\][/tex]
So, the final expanded form of the given logarithmic expression is:
[tex]\[
6 \cdot \log (x) + \log (y) - \log (z)
\][/tex]
