Problem 108 - Algebra and Trigonometry

Question

Without showing the details, explain how to condense \(\ln x-2 \ln (x+1)\)

Step-by-Step Solution

Answer

The condensed form of \(\ln x-2 \ln (x+1)\) is \(\ln \frac{x}{(x+1)^2}\).

Step 1: Apply Logarithm Power Rule

Use the power rule of logarithms to rewrite \(2 \ln (x+1)\) as \(\ln (x+1)^2\). This gives us \(\ln x - \ln (x+1)^2\).

Step 2: Apply Logarithm Subtraction Rule

Use the rule of logarithms that states \(\ln a - \ln b = \ln \frac{a}{b}\) to simplify the expression. This gives us \(\ln \frac{x}{(x+1)^2}\).