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Chapter 4: Problem 19

Solve the following equations for \(x.\) $$e^{2 x}=5$$

Short Answer

Expert verified
x = \frac{\ln(5)}{2}.

Step by step solution

01

- Take the Natural Logarithm of Both Sides

In order to solve the equation, start by taking the natural logarithm (ln) of both sides of the given equation o.e^{2x} = 5.This step helps us bring down the exponent involving x. Therefore, applying the natural logarithm to both sides gives \[\ln(e^{2x}) = \ln(5).\]
02

- Simplify the Left Side

Using the logarithm power rule, \[\ln(e^a) = a,\]we can simplify the left side of the equation. Therefore, \[\ln(e^{2x}) = 2x.\]So now, the equation becomes \[2x = \ln(5).\]
03

- Solve for x

To isolate x, divide both sides of the equation by 2: \[x = \frac{\ln(5)}{2}.\]This gives the solution for x.

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Key Concepts

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

Exponential Functions
Exponential functions are a powerful way to represent a wide range of phenomena, from population growth to radioactive decay. The general form of an exponential function is \[ f(x) = a \times b^x \] where \[a\] is the initial value, \[b\] is the base, and \[x\] is the exponent. In the original problem, the equation is \[ e^{2x} = 5 \], which is an exponential equation with base \[e\], the natural logarithm base approximately equal to 2.718. Exponential functions have unique properties:
  • Their growth rate becomes faster as the value of \[x\] increases.
  • They can be used to model exponential growth or decay situations effectively.
  • The base \[e\] is especially important in natural processes, making it crucial in calculus and differential equations.
Equipped with the understanding of the exponential function, it becomes easier to transform and solve these types of equations using logarithms.
Logarithm Properties
Logarithms are the inverse operations to exponentiation. If you have \[ b^x = y \], the logarithm base \[b\] of \[y\] is \[ \text{log}_b(y) = x \]. In the natural logarithm case, the base is \[e\] and notated as \[ \text{ln}(y) \]. Here are some key logarithm properties that help in manipulating and solving equations:
  • Product Property: \[ \text{log}_b(x \times y) = \text{log}_b(x) + \text{log}_b(y) \]
  • Quotient Property: \[ \text{log}_b\frac{x}{y} = \text{log}_b(x) - \text{log}_b(y) \]
  • Power Property: \[ \text{log}_b(x^a) = a \times \text{log}_b(x) \]
  • Change of Base Formula: \[ \text{log}_b(y) = \frac{\text{log}_k(y)}{\text{log}_k(b)} \]
For the given problem, we used the power property to simplify:
\[ \text{ln}(e^{2x}) = 2x \]. This property allows us to bring down the exponent \[2x\], making the equation solvable using basic algebraic techniques.
Solving Equations
Solving equations, especially exponential ones, often involves isolating the variable to one side. Here's a detailed guide:
  • Taking the Logarithm: When dealing with exponential equations, the first step is generally to take the natural logarithm \[ \text{ln} \] (or any logarithm relevant to the equation) on both sides. For \[ e^{2x} = 5 \], we get \[ \text{ln}(e^{2x}) = \text{ln}(5) \].
  • Using Logarithm Properties: Utilize properties like the power rule to bring the exponent in front. So, \[ \text{ln}(e^{2x}) = 2x \] brings down the \[2x\].
  • Isolating the Variable: Reorganize the equation to solve for \[x\]. In the problem, you do this by dividing both sides by 2, resulting in \[ x = \frac{\text{ln}(5)}{2} \].
  • Finalizing the Solution: Simplify the equation to find the value of \[ x \] clearly.
Understanding these methods will not only help in solving different kinds of exponential equations but also build a solid foundation for more complex algebraic problems.

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