Problem 108 - College Algebra Essentials

Question

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

Step-by-Step Solution

Answer

The condensation of \(\ln x-2 \ln (x+1)\) into a single logarithm expression gives us \(\ln \frac{x}{{(x+1)}^2}\)

Step 1: Apply the power rule

Use the power rule to transform \(2 \ln (x+1)\) into \(\ln {(x+1)}^2\). This gives us a new expression: \( \ln x -\ln {(x+1)}^2\).

Step 2: Apply the quotient rule

Next, utilize the logarithm quotient rule to further simplify the expression. This rule states that the difference of two logarithms is the logarithm of the quotient of their arguments. Thus, we can rewrite the expression as \(\ln \frac{x}{{(x+1)}^2}\)

Step 3: Simplify the expression

Finally, the expression inside the logarithm can be simplified further. This requires finding a common denominator, which is \(x+1\) and reducing the fraction. This leads to the final expression: \( \ln \frac{x}{{(x+1)(x+1)}\)