Problem 46 - A Graphical Approach to College Algebra

Question

Solve each equation. $$e^{3-x}=\left(e^{3}\right)^{-x}$$

Step-by-Step Solution

Answer

The solution is \( x = -\frac{3}{2} \).

Step 1: Understand the given equation

We are given the equation \( e^{3-x} = (e^3)^{-x} \). Our task is to solve for \( x \).

Step 2: Simplify the right-hand side

Notice that the right-hand side, \( (e^3)^{-x} \), can be rewritten using the power rule of exponents. This gives us: \( (e^3)^{-x} = e^{3(-x)} = e^{-3x} \). So, the equation becomes \( e^{3-x} = e^{-3x} \).

Step 3: Equate the exponents

Since the bases are the same (both sides contain \( e \)), we can set the exponents equal to each other: \( 3-x = -3x \).

Step 4: Solve for x

Rearrange the equation \( 3-x = -3x \) to solve for \( x \). Start by adding \( 3x \) to both sides: \( 3 - x + 3x = 0 \). Simplify this to: \( 3 + 2x = 0 \).

Step 5: Isolate x

Subtract 3 from both sides to get \( 2x = -3 \).

Step 6: Solve for x

Divide both sides by 2 to solve for \( x \): \( x = \frac{-3}{2} \).

Key Concepts

Power Rule of Exponents

The power rule of exponents is a fundamental concept in mathematics that helps simplify expressions with exponents. When dealing with an exponent raised to another power, you multiply the exponents. For example, when you have

• a base to an exponent, such as \((a^m)^n\),
• you can rewrite it as \(a^{m \cdot n}\).

In the original exercise, we applied this rule to the right-hand side of the equation:

• \((e^3)^{-x}\) becomes \(e^{3(-x)}\),
• which simplifies to \(e^{-3x}\).

Using this rule makes it easier to work with equations, especially when the purpose is to equate bases and simplify the task of solving for variables like \(x\). Understanding and applying the power rule correctly can significantly reduce complexities in mathematical expressions.

Equating Exponents

Equating exponents is a strategy used when you have two exponential expressions with the same base. If they are equal, their exponents must also be equal. For instance, if you have

• N^a = N^b,
• then necessarily a = b.

This principle operates because the equation implies that raising the same base to two different exponents results in the same value—which is only true if the exponents themselves are the same.
In our original equation \(e^{3-x} = e^{-3x}\), we use this approach to simplify the process:

• With the bases being the same (both are \(e\)), the exponents must equate for both sides to be equal.
• Hence, it leads us to solve the simpler linear equation \(3-x = -3x\).

This method provides a straightforward path to solving for variables in exponential form without having to calculate numerical values for complex such equations.

Solving Linear Equations

Linear equations are solved by finding the value of the unknown variable that makes the equation true. These equations are expressions in the form \(ax + b = 0\), where:

• \(a\) and \(b\) are constants, and
• \(x\) is the variable we solve for.

In the context of our equation \(3-x = -3x\), you start by rearranging terms to isolate the variable \(x\) on one side of the equation:

• Add \(3x\) to both sides: \(3 - x + 3x = 0\),
• this simplifies to \(3 + 2x = 0\).

Next, solve for \(x\):

• Subtract \(3\) from both sides, resulting in \(2x = -3\).
• Finally, divide both sides by \(2\) to obtain \(x = \frac{-3}{2}\).

This sequential approach is key in breaking down the problem to find the variable accurately. Understanding solving techniques for linear equations is critical for solving more complex mathematical problems.