Problem 88 - 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 \(\ln\left(\frac{x}{x^2+2x+1}\right)\).

Step 1: Apply the Power Rule

We start by using the power rule of logs to adjust the second term in the expression, which transforms \(2\ln(x+1)\) into \(\ln((x+1)^2)\). So, the expression becomes: \(\ln x - \ln((x+1)^2)\).

Step 2: Apply the Quotient Rule

Next, the quotient rule of logs is applied, subtracting one log from another is equivalent to dividing the arguments of these logs. The rule transforms \(\ln x - \ln((x+1)^2)\) into \(\ln\left(\frac{x}{(x+1)^2}\right)\).

Step 3: Final Simplification

The expression can be further simplified by expressing \((x+1)^2\) as \(x^2+2x+1\). This simplifies \(\ln\left(\frac{x}{(x+1)^2}\right)\) to \(\ln\left(\frac{x}{x^2+2x+1}\right)\).