Questions: Use properties of logarithms to condense the logarithmic expression. Write the expression as a single logarithm. 4ln x - 5ln x = ln x^4/x^5

Solution Steps

To condense the logarithmic expression, we can use the properties of logarithms. Specifically, we can use the power rule, which states that \( a \cdot \ln(b) = \ln(b^a) \), 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 given in the problem: \[ 4 \ln x - 5 \ln x \]

Using the power rule of logarithms, we can rewrite each term: \[ \ln(x^4) - \ln(x^5) \]

Next, we apply the quotient rule of logarithms, which allows us to combine the two logarithmic terms: \[ \ln\left(\frac{x^4}{x^5}\right) \]

We can simplify the fraction inside the logarithm: \[ \frac{x^4}{x^5} = x^{4-5} = x^{-1} \] Thus, we have: \[ \ln(x^{-1}) \]

Using the property of logarithms, we can express this as: \[

• \ln x \]

Final Answer