To simplify the given logarithmic expression using properties of logarithms, follow these step-by-step transformations:
1. Original Expression:
[tex]\[
4 \ln x - 3 \ln y
\][/tex]
2. Logarithmic Property - Power Rule:
The Power Rule of logarithms states:
[tex]\[
a \ln b = \ln(b^a)
\][/tex]
Using this property:
[tex]\[
4 \ln x = \ln(x^4)
\][/tex]
and
[tex]\[
-3 \ln y = \ln(y^{-3})
\][/tex]
Substituting these back into the expression, we get:
[tex]\[
4 \ln x - 3 \ln y = \ln(x^4) - \ln(y^{-3})
\][/tex]
3. Logarithmic Property - Subtraction Rule:
The Subtraction Rule of logarithms states:
[tex]\[
\ln(a) - \ln(b) = \ln\left(\frac{a}{b}\right)
\][/tex]
Applying this property:
[tex]\[
\ln(x^4) - \ln(y^{-3}) = \ln\left(\frac{x^4}{y^{-3}}\right)
\][/tex]
4. Simplifying the Expression:
Recall that dividing by a negative exponent is equivalent to multiplying by its positive exponent:
[tex]\[
\frac{x^4}{y^{-3}} = x^4 \cdot y^3
\][/tex]
Therefore:
[tex]\[
\ln\left(\frac{x^4}{y^{-3}}\right) = \ln(x^4 y^3)
\][/tex]
Thus, the simplified logarithmic expression is:
[tex]\[
4 \ln x - 3 \ln y = \ln(x^4 y^3)
\][/tex]
So the final answer is:
[tex]\[
\boxed{\ln(x^4 y^3)}
\][/tex]
