Problem 18 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Solve the equation for \(x\). $$(-1.3)^{x-2}=(-1.3)^{2 x+1}$$

Step-by-Step Solution

Answer

The short answer to the problem is: \(x = -3\).

Step 1: Write down the given equation

The given equation is: \((-1.3)^{x-2}=(-1.3)^{2x+1}\)

Step 2: Equate the exponents

Since the bases of both exponentials are equal, we can equate their exponents and write: \(x-2 = 2x+1\)

Step 3: Isolate x and find its value

Now, we can solve for x by isolating it. First, we'll subtract x from both sides of the equation, then subtract 1 from both sides of the equation. \(x - 2 - x = 2x + 1 - x\) \(-2 = x + 1\) Subtracting 1 from both sides, we get: \(-2 - 1 = x\)

Step 4: Simplify and determine the value of x

Finally, we simplify and get the value of x: \(-3 = x\) So, the value of x is -3.

Key Concepts

Solving Exponential Equations

Understanding how to solve exponential equations is key to navigating through various mathematical and real-world problems. An exponential equation is one where the variable appears in the exponent. To tackle these effectively, you'll often start by ensuring that both sides of the equation have the same base. Once this first step is achieved, you can then equate the exponents, because for a given base a, if a^x = a^y, then x must equal y.

Here's a simple guideline to remember:

• Identify the base of the exponentials and check if they are the same.
• If the bases are the same, set the exponents equal to each other.
• Solve the resulting equation to find the value of the unknown variable.

Applying these steps makes what could be a complex problem into something much more manageable.

Equating Exponents

Once you have an exponential equation with the same base on both sides, the next step is equating exponents. This is possible because of a fundamental property of exponents, which states that if a^x = a^y and a is not equal to 1 or 0, then x = y. This allows you to simplify the problem significantly as the complex exponential parts can be removed, leaving you with a more traditional equation.

For example, in our exercise, we were able to set (x-2) and (2x+1) equal to each other because the base (-1.3) was the same for both sides of the equation. This fundamental approach to solving exponential equations is crucial and widely used in algebra.

Isolate the Variable

After simplifying your exponential equation, you’ll often end up with a more familiar linear equation where your goal is to isolate the variable, such as x. Isolating the variable means manipulating the equation so that the variable is on one side by itself.

Typical steps to isolate the variable include:

• Performing the same operation on both sides of the equation to maintain equality.
• Using addition or subtraction to get the variable terms on one side and the constants on the other.
• Applying multiplication or division to solve for the variable when necessary.

The simplicity of these steps hides their importance; mastering them will make solving complex equations much easier.

Mathematical Reasoning

At its heart, solving exponential equations, much like other areas of algebra, relies on mathematical reasoning. This involves logical thinking, pattern recognition, and understanding how different mathematical principles interact with one another. It’s not enough to just apply formulas; a good grasp of how and why certain methods work is essential.

For example, knowing why you can equate exponents when bases are the same and how that simplifies the problem is a result of mathematical reasoning. As you advance in math, developing this logical framework not only helps you solve equations but also to approach problems creatively and efficiently, leading to a deeper understanding of mathematics as a whole.