/*! 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 148 Check each proposed solution by ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 4: Problem 148

Check each proposed solution by direct substitution or with a graphing utility. $$\ln (\ln x)=0$$

Short Answer

Expert verified
The solution to the equation \(\ln (\ln x) = 0\), is \(x \approx 2.71828\) or \(x = e\), where \(e\) is Euler's number. This was confirmed by substituting \(x = e\) back into the original equation and confirming the equality holds.

Step by step solution

01

Understand the equation format

Given the logarithmic equation \(\ln (\ln x) = 0\). The challenge here is to convert this logarithmic form into an equivalent form that can facilitate solving for the value of \(x\). First, we need to understand the idea behind logarithmic form: the equation \(\ln a = b\) can be rewritten in exponential form as \(e^b = a\), where \(e\) denotes Euler's number, approximately equal to 2.71828. This is the base of the natural logarithm.
02

Convert the equation from logarithmic to exponential form

Applying the logic mentioned in Step 1, we can rewrite the equation \(\ln (\ln x) = 0\) in exponential form. We achieve this by setting \(0\) as the exponent of \(e\), giving us \(e^0 = \ln x\). Simplifying \(e^0\) which equals \(1\), we get \(\ln x = 1\). Therefore, the equation simplifies to asking for which \(x\), the natural logarithm of \(x\) equals \(1\).
03

Solve for \(x\)

Next, convert \(\ln x = 1\) back into exponential form to directly solve for \(x\). Applying the same change we did in step 2 gives us: \(e^1 = x\). Thus, \(x = e\). Given the base of the natural logarithm \(e\) is approximately 2.71828, we can conclude that \(x \approx 2.71828\). Check this by direct substitution into the original equation.
04

Check the solution by substituting in the original equation

Our final step involves substituting \(x = e\) into the original logarithmic equation to check that the solution is correct. Substituting \(x = e\) into the equation \(\ln (\ln x) = 0\), we get \(\ln (\ln e) = 0\). \( \ln e = 1\) and thus we have \(\ln 1 = 0\). \(\ln 1 = 0\) is a correct equation. This shows that our solution, \(x = e\), is indeed correct.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Suppose that a population that is growing exponentially increases from \(800,000\) people in 2007 to \(1,000,000\) people in \(2010 .\) Without showing the details, describe how to obtain the exponential growth function that models the data.

Evaluate the indicated logarithmic expressions without using a calculator. a. Evaluate: \(\log _{2} 16\) b. Evaluate: \(\log _{2} 32-\log _{2} 2\) c. What can you conclude about $$\log _{2} 16, \text { or } \log _{2}\left(\frac{32}{2}\right) ?$$

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 \(^{2} ?\)

a. Simplify: \(e^{\ln 3}\) b. Use your simplification from part (a) to rewrite \(3^{x}\) in terms of base \(e\)

One problem with all exponential growth models is that nothing can grow exponentially forever. Describe factors that might limit the size of a population.
See all solutions

Recommended explanations on Math Textbooks

Discrete Mathematics

Read Explanation

Logic and Functions

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Statistics

Read Explanation

Calculus

Read Explanation

Pure Maths

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.