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Chapter 1: Problem 23

Solve for \(x\) without using a calculating utility. $$\ln (1 / x)+\ln \left(2 x^{3}\right)=\ln 3$$

Short Answer

Expert verified
The solution is \( x = \sqrt{\frac{3}{2}} \).

Step by step solution

01

Apply Logarithmic Property

Use the property of logarithms that states \( \ln a + \ln b = \ln (a \times b) \). Apply this to the left side of the equation: \( \ln(1/x) + \ln(2x^3) = \ln((1/x) \times (2x^3)) \). This simplifies to \( \ln(2x^2) \). Thus, the equation becomes \( \ln(2x^2) = \ln 3 \).
02

Remove the Logarithms

Since the natural logarithms on both sides are equal, set the arguments equal: \( 2x^2 = 3 \). This follows from the fact that if \( \ln a = \ln b \), then \( a = b \).
03

Solve for x

Now, solve the equation \( 2x^2 = 3 \) for \( x \). Start by dividing both sides by 2: \( x^2 = \frac{3}{2} \). Then, take the square root of both sides to find \( x \): \( x = \pm \sqrt{\frac{3}{2}} \).
04

Select the Valid Solution

Since \( \ln(1/x) \) requires \( x > 0 \), the negative root is not valid. Therefore, the solution is \( x = \sqrt{\frac{3}{2}} \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Properties of Logarithms
Understanding the properties of logarithms is essential when solving logarithmic equations like the one given. Logarithms have several important properties, but one key property used in this exercise is the product rule:
  • Product Rule: \( \ln a + \ln b = \ln (a \times b) \)
This rule allows the simplification of a sum of two logarithms into the logarithm of a product. In the solution provided, the equation \( \ln(1/x) + \ln(2x^3) \) is simplified to \( \ln((1/x) \times (2x^3)) \). Simplifying further, this becomes \( \ln(2x^2) \).
This step reduces the complexity of the problem significantly, making it easier to solve.
Natural Logarithm
The natural logarithm, denoted as \( \ln \), is a logarithm with base \( e \), where \( e \approx 2.71828 \). Natural logarithms are used extensively in calculus and complex calculations. In this problem, the use of \( \ln \) instead of general logs with another base is particularly important.
Natural logarithms follow the same properties as any logarithm, but they are special because of their base. In the given equation, both sides are expressed in terms of the natural logarithm:
\[ \ln(2x^2) = \ln 3 \]Using the property that if \( \ln a = \ln b \), then \( a = b \), we can rewrite this as \( 2x^2 = 3 \). This shows us how the natural logarithm helps in forming a straightforward equation that we can solve using algebraic methods.
Solving Quadratic Equations
Quadratic equations are equations of the form \( ax^2 + bx + c = 0 \). In this problem, once we eliminate the logarithms, we reduce the equation to the form \( 2x^2 = 3 \). Although this is not quite a standard quadratic form, it can be treated similarly by rearranging terms to solve for \( x \).
Here are key steps:
  • Divide both sides by the coefficient of \( x^2 \), which is 2, getting \( x^2 = \frac{3}{2} \).
  • Take the square root of both sides, considering both the positive and negative possibilities: \( x = \pm \sqrt{\frac{3}{2}} \).
However, given the context of the problem where the logarithm \( \ln(1/x) \) is involved, only positive values of \( x \) are valid. This restricts the solution to \( x = \sqrt{\frac{3}{2}} \). Understanding these steps is crucial for solving not only logarithmic equations but also more complex algebraic forms.

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Most popular questions from this chapter

Give an example of a function \(f\) that is defined on a closed interval, and whose values at the endpoints have opposite signs, but for which the equation \(f(x)=0\) has no solution in the interval.

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Writing Suppose that \(f\) and \(g\) are two functions that are equal except at a finite number of points and that \(a\) denotes a real number. Explain informally why both $$ \lim _{x \rightarrow a} f(x) \quad \text { and } \quad \lim _{x \rightarrow a} g(x) $$ exist and are equal, or why both limits fail to exist. Write a short paragraph that explains the relationship of this result to the use of "algebraic simplification" in the evaluation of a limit.

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The notion of an asymptote can be extended to include curves as well as lines. Specifically, we say that curves \(y=f(x)\) and \(y=g(x)\) are asymptotic as \(x \rightarrow+\infty\) provided $$ \lim _{x \rightarrow+\infty}[f(x)-g(x)]=0 $$ and are asymptotic as \(x \rightarrow-\infty\) provided $$ \lim _{x \rightarrow-\infty}[f(x)-g(x)]=0 $$ In these exercises, determine a simpler function \(g(x)\) such that \(y=f(x)\) is asymptotic to \(y=g(x)\) as \(x \rightarrow+\infty\) or \(x \rightarrow-\infty\) Use a graphing utility to generate the graphs of \(y=f(x)\) and \(y=g(x)\) and identify all vertical asymptotes. $$f(x)=\frac{x^{5}-x^{3}+3}{x^{2}-1}$$
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