Problem 51 - Precalculus with Limits

Question

Using the One-to-One Property In Exercises \(51-54,\) use the One-to-One Property to solve the equation for \(x .\) $$e^{3 x+2}=e^{3}$$

Step-by-Step Solution

Answer

The solution to the equation \(e^{3x+2}= e^3\) is \(x = \frac{1}{3}\).

Step 1: Identify the bases

Here, both sides of the equation are exponential expressions with the same base \(e\). That is \(e^{3x+2}\) and \(e^3\).

Step 2: Apply the One-to-One Property

By the One-to-One Property of exponential functions, when bases are the same, we can set the exponents equal to each other. Therefore, we make the exponents equal: \(3x+2=3\).

Step 3: Solve for \(x\)

The equation can be simplified as: \(3x=3-2\), which simplifies to \(3x=1\). Therefore, dividing both side by 3, \(x = \frac{1}{3}\).

Key Concepts

Exponential Functions

Exponential functions are mathematical functions that involve a variable exponent. They are written in the form of \( f(x) = a^x \), where \( a \) is a constant and \( x \) is the variable exponent. In these functions, the base, \( a \), is typically a positive real number, and it is most common to see exponents that are expressions containing the variable \( x \). For instance, in the equation \( e^{3x+2} = e^3 \), \( e \) is the base, and it is a special number approximately equal to 2.71828, known as Euler's number.

Exponential functions are used to model growth and decay processes, such as population growth, radioactive decay, and interest compounding. They are essential in various scientific computations and financial mathematics.

• They demonstrate rapid changes either increasing (growth) or decreasing (decay).
• Graphs of exponential functions are curves that get steeper or shallower as \( x \) increases or decreases.

Understanding exponential functions is crucial as they frequently appear in real-world applications and more complex mathematical problems.

Solving Equations

Solving equations involves finding the value of a variable that makes the equation true. In the context of exponential equations like \( e^{3x+2} = e^3 \), we use properties like the One-to-One Property to find solutions.

The One-to-One Property states that if two exponential expressions with the same base are equal, then their exponents must also be equal. This is because exponential functions are one-to-one, meaning each input (or exponent) corresponds to one unique output. For the equation given, we see that both sides have the base \( e \), making it simple to apply this property.

• Step 1 is identifying the same base on both sides.
• Step 2 is setting the exponents equal through the One-to-One Property.

This approach simplifies many exponential equations and helps in isolating the variable. Once the equation is reduced to a simple linear equation, basic algebra is applied to solve for \( x \).

Exponent Rules

Exponent rules are guidelines for performing operations involving powers. These rules clarify how to handle bases and exponents in various mathematical contexts. When dealing with exponential equations, these rules become crucial.

• One primary rule is the Product of Powers rule: \( a^m \times a^n = a^{m+n} \).
• Another is the Power of a Power rule: \((a^m)^n = a^{mn}\).

In solving our equation, an implicit understanding of these rules helps deconstruct the equation's components.

When you see an equation like \( e^{3x+2} \), although we don't directly use these rules to solve the equation, knowing how to manipulate exponents arms you with the confidence to solve exponential equations as soon as the bases are made equal. In our specific exercise, setting the terms equal by acknowledging their shared base reflects this understanding of exponentiation. Mastering exponent rules can transform intimidating problems into manageable challenges.