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Chapter 4: Problem 88

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

Short Answer

Expert verified
The condensed form of the expression \(\ln x-2 \ln (x+1)\) is \(\ln\left(\frac{x}{x^2+2x+1}\right)\).

Step by step solution

01

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

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

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

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Most popular questions from this chapter

In Exercises \(1-40,\) use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. $$ \log (10,000 x) $$

In Exercises \(1-40,\) use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. $$ \log _{4}\left(\frac{\sqrt{x}}{64}\right) $$

In Exercises \(71-78,\) use common logarithms or natural logarithms and a calculator to evaluate to four decimal places. $$ \log _{16} 57,2 $$

In Exercises \(41-70\), use properties of logarithms to condense each logarithmic expression. Write the expression as a single logarithm whose coefficient is \(I .\) Where possible, evaluate logarithmic expressions. $$ 5 \ln x-2 \ln y $$

In Exercises \(79-82,\) use a graphing utility and the change-of- base property to graph each function. $$ y=\log _{3} x $$
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