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Chapter 4: Problem 108

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

Short Answer

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

Step by step solution

01

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: \( \ln x -\ln {(x+1)}^2\).
02

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 \(\ln \frac{x}{{(x+1)}^2}\)
03

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: \( \ln \frac{x}{{(x+1)(x+1)}\)

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