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Chapter 3: Problem 104

Solve the logarithmic equation algebraically. Approximate the result to three decimal places. $$\ln (x+1)-\ln (x-2)=\ln x$$

Short Answer

Expert verified
Based on these steps, the solution for the equation can be approximated to three decimal places. The exact values will depend on the solutions obtained from the quadratic equation and the domain validity checks done in the final step.

Step by step solution

01

Use the properties of logarithms

To start off, use the property of logarithms which states that the difference of two logarithms is equal to the logarithm of the quotient. Hence the equation \( \ln (x+1)-\ln (x-2)=\ln x \) becomes \( \ln [(x+1)/(x-2)] = \ln x \).
02

Equate the arguments of the logarithms

Since two logarithms equal to each other have their arguments equivalent, we equate the arguments by setting \( (x+1)/(x-2) = x \).
03

Solve for x

To find the value of x, cross-multiply to get rid of the denominator. Then simplify the equation to obtain a quadratic equation, which can be solved by either factoring, completing the square or using the quadratic formula. For this equation, the quadratic formula can be applied.
04

Find the roots of the quadratic equation

After applying the quadratic formula, we get two roots for x. However, keep in mind that these roots must satisfy the domain restrictions. The values of x must be such that the arguments of the logarithms are always greater than zero. Hence, we must test these roots back into the original equation to ascertain if they fit these conditions.
05

Check the solution

We substitute these roots back into the original equation to check if they satisfy the equation. Only the values which make the original equation valid are the correct solutions. If the value of x yields an undefined logarithm, it is then discarded.

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Most popular questions from this chapter

Solve the equation algebraically. Round the result to three decimal places. Verify your answer using a graphing utility. $$2 x^{2} e^{2 x}+2 x e^{2 x}=0$$

The sales \(S\) (in thousands of units) of a new CD burner after it has been on the market for \(t\) years are modeled by \(S(t)=100\left(1-e^{k t}\right) .\) Fifteen thousand units of the new product were sold the first year. (a) Complete the model by solving for \(k\). (b) Sketch the graph of the model. (c) Use the model to estimate the number of units sold after 5 years.

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