Problem 46 - A Graphical Approach to Precalculus with Limits

Question

Solve each equation. Do not use a calculator. $$e^{3-x}=\left(e^{3}\right)^{-x}$$

Step-by-Step Solution

Answer

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

Step 1: Analyze the Equation Form

The given equation is \( e^{3-x} = (e^3)^{-x} \). Recognize that \( (e^3)^{-x} \) can be rewritten as \( e^{-3x} \) since the property of exponents states \( a^{-b} = \frac{1}{a^b} \). Thus, rewriting the equation, we have \( e^{3-x} = e^{-3x} \).

Step 2: Set the Exponents Equal

Since the bases are the same (\(e\)), we set the exponents equal to one another to solve the equation: \( 3 - x = -3x \).

Step 3: Simplify the Equation

Rearrange the equation \( 3 - x = -3x \) to isolate \( x \). First, add \( 3x \) to both sides: \( 3 - x + 3x = 0 \) which simplifies to \( 3 + 2x = 0 \).

Step 4: Solve for x

Solve the equation \( 3 + 2x = 0 \) for \( x \). Subtract 3 from both sides to get \( 2x = -3 \). Then, divide both sides by 2, giving \( x = -\frac{3}{2} \).

Key Concepts

Exponential Functions

Exponential functions are an essential mathematical concept, characterized by a variable exponent like in the expression \( e^{x} \). In an exponential function, the base remains constant while the exponent varies, which allows these functions to model many real-world phenomena such as population growth, radioactive decay, and interest calculation.

• The general form of an exponential function is \( f(x) = a \, b^{x} \), where \( a \) is a constant coefficient, \( b \) is the base of the exponential, and \( x \) is the variable exponent.
• For the specific case involving the natural exponential function \( e \), which is approximately 2.718, often referred to as Euler's number, many natural logarithms and calculations become simplified.
• In exponential equations, the same base on both sides of an equation allows us to set exponents equal, simplifying the problem significantly.

By understanding the behavior of exponential functions, students can solve equations where exponents are central to the problem. In equations like the given one, \( e^{3-x} \) and \( e^{-3x} \), understanding this allows for straightforward manipulation to solve for unknown variables.

Properties of Exponents

The properties of exponents are rules that simplify calculations and expressions involving powers. These rules are crucial for solving equations like \( e^{3-x} = (e^3)^{-x} \). Knowing these properties allows mathematicians and students to transform and solve equations by manipulating exponents.

• **Power of a Power:** \((a^m)^n = a^{m \cdot n}\). This means raising an exponent to another power multiplies the exponents.
• **Product of Powers:** \(a^m \cdot a^n = a^{m+n}\). When multiplying like bases, simply add the exponents.
• **Quotient of Powers:** \(\frac{a^m}{a^n} = a^{m-n}\). When dividing like bases, subtract the exponents.
• **Negative Exponent:** \(a^{-b} = \frac{1}{a^b}\). This is used to change negative exponents to positive by placing it as a denominator.

In this problem, understanding that \((e^3)^{-x}\) can be written as \(e^{-3x}\) utilizing the negative exponent property allows the problem to be simplified, reducing it down to an algebraic equation with like bases.

Solving Algebraic Equations

Algebraic equations require finding the value of variables that make the equation true. In the context of exponential equations, solving involves reducing the equation to isolate the unknown variable. Consider the equation \( e^{3-x} = e^{-3x} \), where both sides have the same base:

• After expressing both sides with the same base, equate the exponents: \( 3 - x = -3x \).
• Solve the equation through typical algebraic manipulation. An important step includes collecting all terms involving \( x \) onto one side:
• Add \( 3x \) to both sides to make calculation easier, resulting in \( 3 + 2x = 0 \).
• Isolate \( x \) by performing subtraction and then division: subtract 3 to get \( 2x = -3 \), then divide by 2 to solve for \( x = -\frac{3}{2} \).

By systematically applying these rules, one can confidently solve exponential equations, leading to understanding their underlying algebraic structures. Practicing such problems helps reinforce the concepts, making them more intuitive in more complex situations.