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Chapter 4: Problem 26

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

Short Answer

Expert verified
The solution is \( x = e^4 \).

Step by step solution

01

Isolate the logarithmic term

Start by moving the logarithmic term to one side of the equation. Subtract 4 from both sides of the equation: \[4 - \ln{x} = 0 \implies - \ln{x} = -4.\]
02

Solve for the logarithmic term

Next, multiply both sides of the equation by -1 to get: \[ \ln{x} = 4. \]
03

Exponentiate both sides

To get rid of the natural logarithm, exponentiate both sides of the equation. This gives: \[ e^{\ln{x}} = e^4. \] Since \( e^{\ln{x}} = x\), it follows that: \[ x = e^4. \]

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Key Concepts

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

logarithmic equations
Logarithmic equations involve logarithms and often require different strategies for solving. They are equations that contain the logarithmic function. An example is the equation from the exercise:
  • 4 - \( \ln{x} \) = 0
The goal is usually to solve for the unknown variable inside the logarithm. In our example, it's about finding what value of \(x\) makes the equation true.

One common strategy is to isolate the logarithmic term first. This means getting the logarithm by itself on one side of the equation. Once the logarithmic term is isolated, you can proceed to solve for the variable, often using rules of logarithms and exponentiation.
exponentiation
Exponentiation is a mathematical operation that involves raising a base to a certain power. In context of logarithmic equations, exponentiation is often used to cancel out logarithms. For example, if you have \( \ln{x} = 4 \), you can exponentiate both sides:
  • \( e^{\ln{x}} = e^4 \)
The property that \( e^{\ln{x}} = x \) simplifies this to:
  • \( x = e^4 \)
Exponentiation helps to transition from a logarithmic form to a simpler algebraic form, making it easier to solve for the variable.
isolating variables
Isolating the variable is a fundamental step in solving equations. This means getting the variable you are solving for by itself on one side of the equation. In logarithmic equations, it often involves moving terms from one side to the other. For instance, in the equation:
  • 4 - \( \ln x \) = 0
The first step is to isolate the logarithmic term:
  • \( - \ln x \) = -4
This is done by subtracting 4 from both sides. The next step further simplifies this:
  • \( \ln x \) = 4
After isolating the variable, you can then use exponentiation to solve for \( x \), giving the final solution of:
  • \( x = e^4 \)
Isolating variables is a critical technique that helps simplify equations and make them solvable using basic algebraic operations.

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Most popular questions from this chapter

Differentiate. $$y=\ln \frac{(x+1)^{4}}{x-1}$$

\([\text{Hint}:\left.\text { Let } X=2^{x} \text { or } X=3^{x} .\right]\) $$3^{2 x}-12 \cdot 3^{x}+27=0$$

Differentiate. $$y=\ln [\sqrt{x e^{x^{2}+1}}]$$

\([\text{Hint}:\left.\text { Let } X=2^{x} \text { or } X=3^{x} .\right]\) $$2^{2 x}-4 \cdot 2^{x}-32=0$$

The function \(f(x)=(\ln x+1) / x\) has a relative extreme point for \(x>0 .\) Find the coordinates of the point. Is it a relative maximum point?
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