Questions: 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 expression [ 4 ln x-5 ln y ] (4 ln x-5 ln y=) (Simplify your answer.)

Solution Steps

To condense the logarithmic expression \(4 \ln x - 5 \ln y\) into a single logarithm, we can use the properties of logarithms. Specifically, we use the power rule, which states that \(a \ln b = \ln(b^a)\), and the quotient rule, which states that \(\ln a - \ln b = \ln\left(\frac{a}{b}\right)\). First, apply the power rule to each term, and then use the quotient rule to combine them into a single logarithm.

We start with the expression \(4 \ln x - 5 \ln y\). Using the power rule of logarithms, we can rewrite each term: \[ 4 \ln x = \ln(x^4) \quad \text{and} \quad 5 \ln y = \ln(y^5) \] Thus, the expression becomes: \[ \ln(x^4) - \ln(y^5) \]

Next, we apply the quotient rule of logarithms, which states that \(\ln a - \ln b = \ln\left(\frac{a}{b}\right)\). Therefore, we can combine the two logarithmic terms: \[ \ln(x^4) - \ln(y^5) = \ln\left(\frac{x^4}{y^5}\right) \]

Final Answer