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Chapter 4: Problem 24

Solve the following equations for \(x .\) \(e^{x^{2}}=25\)

Short Answer

Expert verified
\ \[ x = \pm \sqrt{\ln(25)} \ \]

Step by step solution

01

Take the Natural Logarithm of Both Sides

To isolate the exponent, take the natural logarithm of both sides of the equation. This gives \ \[ \ln(e^{x^{2}}) = \ln(25) \ \]
02

Simplify Using Properties of Logarithms

Use the property of logarithms \ \[ \ln(e^{a}) = a \ \] \ Simplifying the left-hand side, \ \[ x^{2} = \ln(25) \ \]
03

Solve for x

Take the square root of both sides to solve for \ x \, remembering to consider both the positive and negative roots. This gives \ \[ x = \pm \sqrt{\ln(25)} \ \]

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Key Concepts

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

Exponential Equations
An exponential equation is an equation in which a variable appears in the exponent. In our problem, we have the equation: \( e^{x^{2}} = 25 \). Here, \( e \) is the base of the natural logarithm, and \( x^2 \) is the exponent. Exponential equations can often be solved using logarithms, as logarithms are the inverse operations of exponentiation. This allows us to 'bring down' the exponent and solve for the variable.
Logarithmic Properties
Logarithms have several properties that make them useful for solving equations. A key property is that the natural logarithm of \( e^{a} \) is simply \( a \), or mathematically, \( \ln(e^a) = a \). In our solution, taking the natural logarithm of both sides of the equation \( e^{x^2} = 25 \) gives us: \( \ln(e^{x^2}) = \ln(25) \). Using the property mentioned, this simplifies to \( x^2 = \ln(25) \).
Solving Equations
Once we've simplified our equation using logarithmic properties, the next step is solving the resulting equation \( x^2 = \ln(25) \). To solve for \( x \), we need to isolate the variable. Since \( x \) is squared, we do this by taking the square root of both sides. This gives us: \( x = \pm \sqrt{\ln(25)} \). Remember to include both the positive and negative roots whenever you take a square root.
Square Roots
The square root of a number \( a \), written as \( \sqrt{a} \), is a value \( x \) such that \( x^2 = a \). For any positive number \( a \), there are two square roots: \( \sqrt{a} \) and \( -\sqrt{a} \). In our equation, \( x = \pm \sqrt{\ln(25)} \), we need to consider both the positive and negative roots because both values satisfy the original equation \( e^{x^2} = 25 \). So, the solution to the problem is not just one value but two: \( x = \sqrt{\ln(25)} \) and \( x = -\sqrt{\ln(25)} \).

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