Problem 28 - College Algebra

Question

For the following exercises, use logarithms to solve. $$ 3 e^{3-3 x}+6=-31 $$

Step-by-Step Solution

Answer

No solution possible; the equation implies an impossible situation.

Step 1: Isolate the Exponential Term

The first step is to move the constant term on the left side to the right side of the equation. We have the equation:\[3 e^{3-3x} + 6 = -31\]Subtract 6 from both sides:\[3 e^{3-3x} = -31 - 6 \]\[3 e^{3-3x} = -37\]

Step 2: Divide by the Coefficient of the Exponential Term

Now, we need to isolate the exponential expression by dividing both sides by 3.\[e^{3-3x} = \frac{-37}{3}\]

Step 3: Check Feasibility

At this point, we should recognize that there is an error. The exponential function \(e^{3-3x}\) is always positive, as \(e^y > 0\) for any real number \(y\). An equation \(e^{3-3x} = \frac{-37}{3}\) suggests an impossible situation because it implies a positive number equals a negative number, which cannot occur.

Key Concepts

Exponential Functions

Exponential functions are a critical concept in mathematics, characterized by the constant base and a variable exponent. For example, in the function \( e^{3-3x} \), the base \(e\) is the Euler's number, approximately equal to 2.718, and the exponent is \(3-3x\). These functions have unique properties:

• Exponential growth or decay: This depends on the sign of the exponent. A positive exponent usually leads to growth, while a negative exponent indicates decay.
• Always positive: No matter the exponent, \( e^y \) will always be greater than zero.
• Inverses of logarithmic functions: Exponential functions are natural inverses of logarithms, which means understanding one helps in understanding the other.

In our problem, the exponential equations reflect a scenario where solving for \(x\) directly through algebraic means requires certain manipulations to isolate variables and make logical conclusions based on the nature of these functions.

Feasibility Analysis

Feasibility analysis in mathematics involves verifying whether a proposed solution scenario is possible or valid. During the process of solving equations, especially non-linear ones such as exponential functions, it's vital to ensure that the resulting expressions stand up to mathematical scrutiny. In our exercise, we arrived at an equation \[e^{3-3x} = \frac{-37}{3}\] Here, feasibility analysis helps us recognize that exponential functions can never equal a negative number.

• Exponential expressions always result in positive values.
• A negative result implies a contradiction.

Recognizing the impossibility of the situation saved us from further incorrect calculations and showcases the importance of feasibility in mathematical problem-solving.

Algebraic Manipulation

Algebraic manipulation refers to the various techniques used to rearrange and simplify equations to isolate variables, solve equations, or simplify expressions. In solving the given exercise, these techniques were essential in understanding the progression towards the final result.Let's see some core techniques used:

• Isolating variables: To simplify the exponential equation, we shifted terms across the equal sign and adjusted constants, as seen with the subtraction of 6 in the initial steps.
• Dividing to simplify: After isolating the exponential term \(3e^{3-3x}\), dividing both sides of the equation resulted in \(e^{3-3x} = \frac{-37}{3}\).
• Recognizing infeasibility: Algebraic manipulation isn't just about numbers; it's about logic. Identifying the negative result when the quotient was resolved highlighted the critical realization of an impossibility.

Algebraic manipulation is foundational in tackling such problems, allowing us to transform complex equations into clear, comprehensible steps.