/*! 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 71 Solve the inequality for \(x\). ... [FREE SOLUTION] | 91影视

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Chapter 1: Problem 71

Solve the inequality for \(x\). $$ -2<\ln x<0 $$

Short Answer

Expert verified
The solution for x is that it should be more than 0.135 and less than 1, so (\(0.135, 1\))

Step by step solution

01

Understand the transformation of ln to exponential form

We have a logarithmic inequality of the type \(-2 < \ln x < 0\). To solve this inequality, we first need to change the logarithmic form into exponential form. From math properties we know that \(y = \ln x\) is equivalent to \(x = e^y\). Here, \(y = -2\) and \(y = 0\) on left and right side correspondingly of inequality.
02

Apply transformations

We can now transform our inequality from logarithmic form into exponential form. Doing this, we get : \(e^{-2} < x < e^{0}\)
03

Solve inequality

Now, calculate the exponential expressions. This gives: \(0.135 < x < 1\)

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Key Concepts

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

Logarithmic Inequalities
Logarithmic inequalities involve expressions with a logarithm, such as \(-2 < \ln x < 0\). When solving these, our goal is to find the range of values for \(x\) that satisfy the inequality.

Logs help us handle exponential relationships more easily, showing how many times one number must be multiplied by itself to reach another number. The inequality here tells us that \(\ln x\) is greater than \(-2\) and less than \(0\). This means we are looking at numbers that, when put inside a logarithm, give values within this range.

To make sense of log inequalities, we translate them into something more straightforward by using exponential forms.
Exponential Form
Converting logarithmic inequalities into exponential form makes them easier to manage. The transition uses the property that if \(y = \ln x\), then \(x = e^y\). Here, \(e\) is a mathematical constant approximately equal to \(2.71828\).

In our inequality, \(-2 < \ln x < 0\), replacing \(\ln x\) with \(y\) gives us two parts: \(-2 < y < 0\).
  • The left side, \(-2 < y\), converts to \(e^{-2} < x\).
  • The right side, \(y < 0\), converts to \(x < e^{0}\).
So the exponential form is \(e^{-2} < x < e^{0}\), which means \(x\) is greater than \(e^{-2}\) and less than \(e^0\).
Solving Inequalities
To solve our transformed inequality, we need to calculate the exponential parts. This means finding the values of \(e^{-2}\) and \(e^0\).

It's helpful to know that \(e^0 = 1\) since any number raised to the power of zero is one. The value of \(e^{-2}\) is slightly less straightforward, but it approximately equals \(0.135\).

So the solution to the inequality \(e^{-2} < x < e^{0}\) becomes \(0.135 < x < 1\). This tells us that \(x\) must be a number between \(0.135\) and \(1\) to satisfy the original inequality.
Mathematical Transformations
Transformations help us change expressions into forms that are easier to work with. By transitioning from a logarithmic to an exponential form, we simplify the inequality process.

This strategy is commonly used in solving logarithmic inequalities. It makes the problem more tractable and allows us to use straightforward arithmetic rather than complex manipulation of logs.

Understanding these transformations is key to mastering inequalities. They enable us to navigate between different mathematical expressions, offering clearer insights into the relationships between numbers and their logarithmic or exponential counterparts.

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Most popular questions from this chapter

Use the Intermediate Value Theorem to show that for all spheres with radii in the interval [1,5] , there is one with a volume of 275 cubic centimeters.

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