Certainly! Let's expand the given logarithmic expression [tex]\(\ln \left(\frac{4 y - 7}{y}\right)\)[/tex] using the properties of logarithms, specifically the quotient rule.
Step-by-Step Solution:
1. Identify the quotient inside the logarithm:
We have the expression [tex]\(\ln \left(\frac{4 y - 7}{y}\right)\)[/tex].
2. Apply the quotient rule of logarithms:
The quotient rule states that [tex]\(\ln \left(\frac{a}{b}\right) = \ln (a) - \ln (b)\)[/tex], where [tex]\(a\)[/tex] and [tex]\(b\)[/tex] are positive real numbers.
Here, [tex]\(a = 4 y - 7\)[/tex] and [tex]\(b = y\)[/tex].
3. Expand the given logarithmic expression:
Using the quotient rule, we can write:
[tex]\[
\ln \left(\frac{4 y - 7}{y}\right) = \ln (4 y - 7) - \ln (y)
\][/tex]
4. Analyze the simplified result:
However, we can see that there may be a more concise and elegant form.
5. Refine the derived expression:
Notice that inside the logarithm, we have the fraction [tex]\(\frac{4 y - 7}{y}\)[/tex]. We can rewrite this fraction as a difference:
[tex]\[
\frac{4 y - 7}{y} = 4 - \frac{7}{y}
\][/tex]
6. Substitute the refined fraction back into the logarithm:
Thus, our logarithm becomes:
[tex]\[
\ln \left(\frac{4 y - 7}{y}\right) = \ln \left(4 - \frac{7}{y}\right)
\][/tex]
So, the expanded expression using the quotient rule and further simplification is:
[tex]\[
\ln \left(4 - \frac{7}{y}\right)
\][/tex]
