/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 57 Solve the logarithmic equation a... [FREE SOLUTION] | 91影视

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Chapter 3: Problem 57

Solve the logarithmic equation algebraically. Approximate the result to three decimal places. $$\ln (x+5)=\ln (x-1)-\ln (x+1)$$

Short Answer

Expert verified
The solution to the equation is \(x = 1.667\) to three decimal places.

Step by step solution

01

Combine the Logarithms on the Right Hand Side

According to the law of logarithms, the logarithm of a quotient can be written as the difference of the logarithms. Therefore, we rewrite the right-hand side of the equation as follows: \[\ln \left(\frac{x-1}{x+1}\right)\]
02

Equate the Logarithmic Expressions

Next, we can set the two logarithmic expressions equal to each other: \[\ln(x+5) = \ln\left(\frac{x-1}{x+1}\right)\]
03

Solve for x

To solve for x, since we have two equivalent logarithmic expressions, we can equate the arguments and solve for x: \[(x+5) = \left(\frac{x-1}{x+1}\right)\]Solving this equation will involve clearing the fraction on the right-hand side and rearranging to isolate x.
04

Clear the Fraction and Rearrange the Equation

Cross-multiply equation to clear the fraction and rearrange to isolate x: \[x(x + 1) = (x + 5)(x - 1)\]Expanding both sides of the equation gives:\[x^2 + x = x^2 + 4x - 5\]Rearranging terms and simplifying gives us the equation:\[3x = 5\]Solving this equation gives us the value of x.
05

Solve for x

Finally, to find x, divide both sides of the equation by 3: \[x = \frac{5}{3}\]Therefore, x = 1.667 when rounded to three decimal places.

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Most popular questions from this chapter

Using the One-to-One Property In Exercises \(73-76,\) use the One-to-One Property to solve the equation for \(x\). \(\ln \left(x^{2}-x\right)=\ln 6\)

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