/*! 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 30 Solve the following equations fo... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 4: Problem 30

Solve the following equations for \(x .\) \(2 \ln x=7\)

Short Answer

Expert verified
x = e^{3.5}

Step by step solution

01

- Isolate the logarithmic function

The given equation is \(2 \, \text{ln} \, x = 7\). To isolate the logarithmic function, divide both sides by 2: \( \text{ln} \, x = \frac{7}{2} \).
02

- Exponentiate both sides

To solve for \(x\), exponentiate both sides to get rid of the natural logarithm. Recall that \(e^{\text{ln} \, x} = x\). Hence, \(x = e^{\frac{7}{2}}\).
03

- Simplify

Simplify the expression to find the value of \(x\). Thus, \(x = e^{3.5}\).

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!

Key Concepts

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

natural logarithm
The natural logarithm, often written as \(\text{ln}\), is the logarithm with the base \(e\), where \(e\) is approximately equal to 2.71828. The natural logarithm of a number is the power to which \(e\) must be raised to obtain that number.
In the equation \(2 \ln x = 7\), \( \ln x \) represents the natural logarithm of \(x\). This function is useful for solving equations involving exponential growth or decay. When isolating the logarithm, it鈥檚 crucial to remember that the inverse operation of a natural logarithm is exponentiation by \(e\).
exponentiation
Exponentiation involves raising a number (the base) to the power of an exponent. In our example, once we have isolated \( \ln x \) by dividing both sides by 2, we need to undo the logarithm to solve for \(x\).
Since \( \ln x \) is the exponent to which \(e\) must be raised to obtain \(x\), exponentiating both sides helps. Here鈥檚 how it's done:
We know that if \( \ln x = \frac{7}{2} \), then \( e^{\ln x} = e^{\frac{7}{2}} \). Using the property \( e^{\ln x} = x \), we deduce that \( x = e^{\frac{7}{2}} \).
This technique transforms the equation from a logarithmic form to an exponential form, making it easier to solve.
isolating variables
Isolating the variable is crucial in solving any equation as it helps to simplify the process. For our equation \(2 \ln x = 7\), we want to isolate the \(\text{ln} x \) term.
We accomplish this by dividing both sides of the equation by 2:
\[ \ln x = \frac{7}{2} \].

With the logarithm isolated on one side, it becomes easier to apply subsequent mathematical operations like exponentiation to solve for \(x\). Isolating variables often serves as the foundation for simplified and effective problem-solving.
simplification
Simplification is the final step to arrive at the most straightforward form of the answer. Once we have expressed \(x\) in terms of exponentiation, we need to simplify our result.
From the previous steps, we have \( x = e^{3.5} \). This is already simplified to its exponential form.

In summary, simplifying an equation means breaking down the mathematical expressions into their simplest form. For this type of problem, where the result is in exponential form, we usually leave the expression as \( e^{3.5} \) unless a decimal approximation is specifically required.

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

(a) Find the point on the graph of \(y=e^{-x}\) where the tangent line has slope \(-2\). (b) Plot the graphs of \(y=e^{-x}\) and the tangent line in part (a).

Find \(k\) such that \(2^{x}=e^{k x}\) for all \(x\).

Solve the following equations for \(x .\) \(4-\ln x=0\)

Solve the following equations for \(x .\) \(\ln (\ln 3 x)=0\)

Differentiate the following functions. \(y=\ln (k x), k\) constant
See all solutions

Recommended explanations on Math Textbooks

Calculus

Read Explanation

Logic and Functions

Read Explanation

Geometry

Read Explanation

Probability and Statistics

Read Explanation

Decision Maths

Read Explanation

Mechanics 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.