Problem 108 - College Algebra

Question

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

Step-by-Step Solution

Answer

The condensed form of the expression \(\ln x-2 \ln (x+1)\) is \(\ln (x/(x+1)^2)\

Step 1: Apply the multiplication property of Logarithms

Applying the multiplication property of logarithms which states \( a \ln b = \ln (b^a) \) to the second term \(-2 \ln (x+1)\), it turns into \(- \ln (x+1)^2\)

Step 2: Apply the division property of Logarithms

The expression now becomes \( \ln x - \ln (x+1)^2\). Now, apply the division property of logarithms that states \( \ln a - \ln b = \ln (a/b) \) to this expression, it simplifies to \(\ln (x/(x+1)^2)\)

Step 3: Final Answer

The condensed form of the expression \(\ln x-2 \ln (x+1)\) is thus \(\ln (x/(x+1)^2)\)