Use properties of logarithms to condense the logarithmic expression below. Write the expression as a single logarithm whose coefficient is 1. Where possible, evaluate logarithmic expressions.

[tex]\[ 4 \ln x - 3 \ln y = \square \][/tex]

(Simplify your answer.)



Answer :

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]