To solve the expression [tex]\(\log \frac{x}{y^5}\)[/tex], we need to apply the properties of logarithms. Here’s a detailed, step-by-step solution:
1. Understand the Expression:
We need to take the logarithm of the fraction [tex]\(\frac{x}{y^5}\)[/tex].
2. Logarithm of a Quotient:
One of the key properties of logarithms is that the logarithm of a quotient is equal to the difference of the logarithms. In mathematical terms:
[tex]\[
\log \left(\frac{a}{b}\right) = \log (a) - \log (b)
\][/tex]
Here, [tex]\(a = x\)[/tex] and [tex]\(b = y^5\)[/tex], so:
[tex]\[
\log \left(\frac{x}{y^5}\right) = \log (x) - \log (y^5)
\][/tex]
3. Logarithm of a Power:
Another useful property of logarithms is that the logarithm of a power can be expressed as the exponent times the logarithm of the base. In mathematical terms:
[tex]\[
\log (a^b) = b \log (a)
\][/tex]
Here, [tex]\(a = y\)[/tex] and [tex]\(b = 5\)[/tex], so:
[tex]\[
\log (y^5) = 5 \log (y)
\][/tex]
4. Substitute the Logarithm of the Power:
Substitute [tex]\(\log (y^5)\)[/tex] back into our expression:
[tex]\[
\log \left(\frac{x}{y^5}\right) = \log (x) - 5 \log (y)
\][/tex]
Therefore, the simplified expression for [tex]\(\log \frac{x}{y^5}\)[/tex] is:
[tex]\[
\log \left(\frac{x}{y^5}\right) = \log (x) - 5 \log (y)
\][/tex]
