/*! 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 99 Solve each equation. $$\ln (2 ... [FREE SOLUTION] | 91影视

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Chapter 4: Problem 99

Solve each equation. $$\ln (2 x+1)+\ln (x-3)-2 \ln x=0$$

Short Answer

Expert verified
The solution for the given equation is \( x = \frac{3 + \sqrt{5}}{2} \).

Step by step solution

01

Simplify the equation using the properties of logarithms

There are few basic properties of logarithms that can be employed here: The first property is \( \ln a + \ln b = \ln (ab) \). Using this property, the first two terms of the equation can be combined as: \( \ln (2 x+1)+\ln (x-3) = \ln[(2x+1)(x-3)] \) The second property is \( n \cdot \ln a = \ln( a^n ) \). Using this property, the last term can be rewritten as: \( -2 \ln x = \ln (x^{-2}) \) Now, replace these transformed terms back in the original equation, we get: \( \ln[(2x+1)(x-3)] + \ln (x^{-2}) = 0 \)
02

Combine the logarithms into a single expression

Again use the first property of logarithm i.e., \( \ln a + \ln b = \ln (ab) \) to combine the obtained equation: \( \ln[(2x+1)(x-3)(x^{-2})] = 0 \)
03

Convert the log equation into an exponential equation

We know that \( \ln a = 0 \) if and only if \( a = 1 \). So let's set the expression inside the logarithm equal to one: \( (2x+1)(x-3)(x^{-2}) = 1 \)
04

Solve the resulting equation

Solving this equation leads us to the final step. Expanding and rearranging gives: \( 2x (x^{-2}) - 3(x^{-2}) + x^{-1} - 1 = 0 \) Simplifying this gives us: \( 2 - \frac{3}{x} + \frac{1}{x^2} - 1 = 0 \) Multiply all terms by \( x^2 \) to get rid of denominators: \( 2x^2 - 3x + 1 - x^2 = 0 \), which simplifies to: \( x^2 - 3x + 1 = 0 \) This is a quadratic equation, which can be solved using the quadratic formula: \( x = \frac{-(-3) \pm \sqrt{(-3)^2 - 4 * 1 * 1}}{2 * 1} \) thus giving us the two solutions: \( x = \frac{3 \pm \sqrt{5}}{2} \)
05

Reject extraneous solutions

Sometimes solving these equations yield solutions that are not valid in the original equation, because they might lead to the log of a negative number or zero. Checking the obtained solutions:\( x = \frac{3 - \sqrt{5}}{2} \) eventually gives us a log of negative number when plugging back into the original equation, so this solution is discarded.\( x = \frac{3 + \sqrt{5}}{2} \) is valid and does not make the arguments of any log negative, and so it is the single solution for this equation.

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Most popular questions from this chapter

The loudness level of a sound, \(D,\) in decibels, is given by the formula $$D=10 \log \left(10^{12} I\right)$$ where \(I\) is the intensity of the sound, in watts per meter \(^{2} .\) Decibel levels range from \(0,\) a barely audible sound, to \(160,\) a sound resulting in a ruptured eardrum. (Any exposure to sounds of I3 0 decibels or higher puts a person at immediate risk for hearing damage.) What is the decibel level of a normal conversation, \(3.2 \times 10^{-6}\) watt per meter \(^{2} ?\)

Describe the following property using words: \(\log _{b} b^{x}=x\).

Research applications of logarithmic functions as mathematical models and plan a seminar based on your group's research. Each group member should research one of the following areas or any other area of interest: \(\mathrm{pH}\) (acidity of solutions), intensity of sound (decibels), brightness of stars, human memory, progress over time in a sport, profit over time. For the area that you select, explain how logarithmic functions are used and provide examples.

Students in a psychology class took a final examination. As part of an experiment to see how much of the course content they remembered over time, they took equivalent forms of the exam in monthly intervals thereafter. The average score for the group, \(f(t),\) after \(t\) months was modeled by the function $$f(t)=88-15 \ln (t+1), \quad 0 \leq t \leq 12$$ a. What was the average score on the original exam? b. What was the average score after 2 months? 4 months? 6 months? 8 months? 10 months? one year? c. Sketch the graph of \(f\) (either by hand or with a graphing utility). Describe what the graph indicates in terms of the material retained by the students.

Graph \(f\) and \(g\) in the same viewing rectangle. Then describe the relationship of the graph of g to the graph of \(f\). $$f(x)=\ln x, g(x)=\ln (x+3)$$
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