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Chapter 4: Problem 13

Solve each equation for \(x\). \(e^{5 x}=e^{20}\)

Short Answer

Expert verified
x = 4

Step by step solution

01

Understand the equation

The given equation is an exponential equation with the same base on both sides: \[ e^{5 x} = e^{20} \]
02

Set the exponents equal to each other

Since the bases are the same, set the exponents equal to each other:\[ 5x = 20 \]
03

Solve for x

Divide both sides by 5 to solve for \( x \):\[ x = \frac{20}{5} \]Simplify the fraction:\[ x = 4 \]

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Key Concepts

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

solving exponential equations
To solve exponential equations, you often need to make the formats on both sides of the equation the same. This can involve getting the same base for the exponents. Once the bases are the same, you can set the exponents equal to each other and solve for the unknown variable.

Let's break this down:
  • Recognize that you are dealing with an exponential equation. If the bases are the same on both sides, like our example with base \(e\), you can proceed by setting the exponents equal.
  • Then, solve the simple algebraic equation that results from equating the exponents.
In our specific example, we equated the exponents because both sides of the equation have the base \( e \). This transformed the equation into a straightforward linear equation, \( 5x = 20 \).
properties of exponents
Understanding the properties of exponents is crucial when dealing with exponential equations. Key properties include:
  • When bases are the same, you can set their exponents equal. For instance, if \( a^{m} = a^{n} \), then \( m = n \).
  • \

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