Question

Write the expression as the logarithm of a single quantity. $$ 3 \ln x+2 \ln y-4 \ln z $$

Short answer

\(\ln \frac{x^3 y^2}{z^4}\)

Step-by-step solution

Use the Property of Logarithms

Starting with \(3 \ln x + 2 \ln y - 4 \ln z\), use the property of logarithms \(a \ln b = \ln b^a\) to rewrite the expression as \(\ln x^3 + \ln y^2 - \ln z^4\)

Apply Logarithm Addition Property

Use the property that \( \ln a + \ln b = \ln ab \) to combine the first two terms, giving \(\ln (x^3 y^2) - \ln z^4\)

Use Logarithm Subtraction Property

Use the property that \( \ln a - \ln b = \ln \frac{a}{b} \) to combine the remaining terms, giving the final form \(\ln \frac{x^3 y^2}{z^4}\)