/*! 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 65 Solve each logarithmic equation ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 4: Problem 65

Solve each logarithmic equation in Exercises \(49-92 .\) Be sure to reject any value of \(x\) that is not in the domain of the original logarithmic expressions. Give the exact answer. Then, where necessary, use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. $$ 6+2 \ln x=5 $$

Short Answer

Expert verified
The solution to the equation \(6+2 \ln x=5\) is \(x = e^{-1/2}\) which is approximately 0.61 when rounded to two decimal places.

Step by step solution

01

Isolate the logarithm

First, we have to isolate the logarithmic term. So, start by subtracting 6 from both sides of the equation: \(2 \ln x = 5 - 6\) which simplifies to \(2 \ln x = -1\)
02

Get rid of coefficient of logarithm

Next, divide both sides by 2 to get rid of the 2 that is multiplied with the logarithm. This results into: \(\ln x = -1/2\)
03

Convert the logarithmic equation into an exponential equation

We then rewrite the equation in exponential form. Remember that \(\ln x\) is log base \(e\) of \(x\), so the equation becomes: \(e^{-1/2} = x\)
04

Check and round off the solution

The solution must be within the domain of the original equation, i.e., \(x > 0\). If \(x = e^{-1/2}\) is positive, it's a valid solution. Calculate the decimal value of \(x\) and round it off to two decimal places if needed.

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

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. $$\frac{\log _{2} 8}{\log _{2} 4}=\frac{8}{4}$$

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.

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

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 130 decibels or higher puts a person at immediate risk for hearing damage.) Use the formula to solve. The sound of a blue whale can be heard 500 miles away, reaching an intensity of \(6.3 \times 10^{6}\) watts per meter\(^{2}\). Determine the decibel level of this sound. At close range, can the sound of a blue whale rupture the human eardrum?

Use your graphing utility to graph each side of the equation in the same viewing rectangle. Then use the \(x\) -coordinate of the intersection point to find the equation's solution set Verify this value by direct substitution into the equation. $$ 5^{x}=3 x+4 $$
See all solutions

Recommended explanations on Math Textbooks

Mechanics Maths

Read Explanation

Statistics

Read Explanation

Applied Mathematics

Read Explanation

Geometry

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.