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Chapter 4: Problem 26

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

Short Answer

Expert verified
\[ x = e^4 \]

Step by step solution

01

Isolate the logarithmic term

Subtract 4 from both sides of the equation \[ 4 - \ln x = 0 \] becomes \[ - \ln x = -4 \]
02

Simplify the equation

Multiply both sides of the equation by -1 to get rid of the negative sign. \[ \ln x = 4 \]
03

Exponentiate both sides

To solve for x, exponentiate both sides using base e (the natural logarithm base). \[ e^{ \ln x} = e^{4} \] which simplifies to \[ x = e^{4} \]

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Key Concepts

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

Natural Logarithm
The natural logarithm, denoted as \(\text{ln}\), is a logarithm with the base \(e\). Here, \(e\) is an irrational and transcendental number approximately equal to 2.71828. The main purpose of natural logarithms is to reverse the process of exponentiation involving \(e\). For example, if \(e^{4}\) is a value, taking the natural logarithm of it, \(\text{ln}(e^{4})\), will give us back the exponent, which is 4.
In the exercise, we start with an equation involving the natural logarithm: \[4 - \text{ln}(x) = 0\]. To solve it, you need to isolate the \(\text{ln}(x)\) term by performing basic algebraic operations.
Exponentiation
Exponentiation is the process of raising a number to a power. The inverse operation of exponentiation is taking a logarithm. If you have an equation like \(\text{ln}(x) = 4\), the way to solve for \(x\) is to exponentiate both sides using the base \(e\).
So, in the exercise, after isolating \(\text{ln}(x)\) to get \(\text{ln}(x) = 4\), you exponentiate both sides, yielding \((e^{\text{ln}(x)} = e^{4})\).
Since exponentiating a natural logarithm essentially 'cancels' out the \(\text{ln}\), the solution simplifies to \(x = e^{4}\).
Isolation of Variables
Isolating variables is a fundamental algebraic technique used to solve equations. The goal is to get the variable you are solving for on one side of the equation by itself. In the exercise, we start with the equation \(\text{4 - ln(x)} = 0\).
First, subtract 4 from both sides to isolate the logarithmic term, resulting in \(\text{-ln(x)} = -4\).
Then, multiply both sides by -1 to obtain a positive \(\text{ln}(x) = 4\).
This step prepares the equation for further simplification and solving.
Base e
The base \(e\), approximately equal to 2.71828, is used as the base for natural logarithms and plays a crucial role in various areas of mathematics, especially in calculus. It provides a natural way to describe growth and decay processes. In the context of solving logarithmic equations, \(e\) is particularly useful because it simplifies the inverse operation.
For the exercise: exponentiating both sides of \(\text{ln}(x) = 4\) using base \(e\) yields \(e^{\text{ln}(x)} = e^{4}\), simplifying immediately to \(x = e^{4}\). This method makes solving the equation straightforward and highlights the special properties of base \(e\).

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Most popular questions from this chapter

Solve the given equation for \(x .\) \(2(\ln x)^{2}+\ln x-1=0\)

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