Problem 91 Solve using any method. $$\ln ... [FREE SOLUTION]

91影视

Precalculus: Graphs and Models, A Right Triangle Approach

Marvin L. Bittinger, Judith A. Beecher, David J. Ellenbogen

$Math Studyset 91影视 Explanations$ Math

6 Edition

Chapter 5: Problem 91

Solve using any method. $$\ln x^{\ln x}=4$$

Short Answer

Expert verified

The solutions are $ x = e^2 $ and $ x = e^{-2} $.

Step by step solution

Simplify the Equation

Given the equation $ \ln x^{\ln x} = 4 $, we use the property of logarithms that $ \ln a^b = b \ln a $. Therefore, our equation becomes: \[ (\ln x) \cdot (\ln x) = 4 \] or more simply: \[ (\ln x)^2 = 4 \]

Solve for $ \ln x $

Next, take the square root of both sides to solve for $ \ln x $: \[ \ln x = \pm \sqrt{4} \] Simplifying further yields: \[ \ln x = \pm 2 \]

Solve for $ x $

Now, we solve for $ x $ by exponentiating both sides: \[ \text{If } \ln x = 2, \quad x = e^2 \] \[ \text{If } \ln x = -2, \quad x = e^{-2} \] Therefore, the solutions are: \[ x = e^2 \quad \text{or} \quad x = e^{-2} \]

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.

Logarithm Properties

Understanding the properties of logarithms is essential when solving logarithmic equations. One important property is the power rule, which states that $\text{ln}(a^b) = b \text{ln}(a)$. This property is used to simplify expressions involving exponents inside logarithms.

For example, in the original exercise, the equation $\text{ln}(x^{\text{ln} x}) = 4$ was given. By applying the power rule, we simplified it to $(\text{ln} x) \times (\text{ln} x) = \text{ln}(x)^2 = 4$.

In general, two key properties to remember are:

\text{ln}(a \times b) = \text{ln}(a) + \text{ln}(b)
\text{ln}\bigg(\frac{a}{b}\bigg) = \text{ln}(a) - \text{ln}(b)

These properties allow us to break down complex logarithmic expressions into more manageable parts.

Exponentiation

Exponentiation is the process of raising a number to a certain power. In the logarithmic equation context, exponentiation is used to 'reverse' the logarithm operation.

In the exercise, after finding that $\text{ln} x = \text{卤} 2$, we needed to find the value of $x$. To do this, we exponentiated both sides of the equation using the base $e$, which is the natural logarithm's base:

If $\text{ln} x = 2$, then $x = e^2$
If $\text{ln} x = -2$, then $x = e^{-2}$.

Exponentiation is crucial for converting logarithmic equations back into their exponential form, allowing us to solve for the variable.

Quadratic Equations

In the given problem, we transformed the logarithmic equation into a quadratic equation. A quadratic equation is generally in the form $ax^2 + bx + c = 0$ and can be solved using various methods like factoring, the quadratic formula, or completing the square.

For the given exercise, the equation $(\text{ln} x)^2 = 4$ is already in the standard quadratic form $u^2 = 4$, where $u = \text{ln} x$. This simplifies to two solutions:

$\text{ln} x = 2$
$\text{ln} x = -2$

By solving these simple quadratic forms, we found that the possible values for $x$ are $e^2$ and $e^{-2}$. Quadratic equations often appear in logarithmic problems when we square terms or simplify complex expressions.

One App. One Place for Learning.

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

Get started for free

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

91影视

Solve using any method. $$\ln x^{\ln x}=4$$

Short Answer

Step by step solution

Simplify the Equation

Solve for \( \ln x \)

Solve for \( x \)

Key Concepts

Logarithm Properties

Exponentiation

Quadratic Equations

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Geometry

Decision Maths

Mechanics Maths

Statistics

Applied Mathematics

Probability and Statistics

Study anywhere. Anytime. Across all devices.