Problem 13 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Solve the equation for \(x\). $$6^{2 x}=6^{4}$$

Step-by-Step Solution

Answer

Since the bases are the same, we can set the exponents equal to each other: \(2x = 4\). To solve for x, divide both sides by 2, resulting in \(x = 2\). So, the solution for the equation \(6^{2x} = 6^4\) is \(x = 2\).

Step 1: Write down the equation

We are given the equation \(6^{2x} = 6^{4}\).

Step 2: Set the exponents equal to each other

Since the bases are the same (6), we can set the exponents equal to each other: \(2x = 4\).

Step 3: Solve for x

Now we will solve for x in the equation \(2x = 4\). To do this, divide both sides by 2: \(x = 2\).

Step 4: State the solution

The solution for the equation \(6^{2x} = 6^4\) is \(x = 2\).

Key Concepts

Solving Equations

When solving equations, our goal is to find the value of the variable that satisfies the equation. We start by understanding the structure of the given equation. Here, we have an exponential equation: \(6^{2x} = 6^{4}\). The key is to simplify the equation step-by-step.

• Identify common bases or factors.
• Isolate the variable you're solving for.
• Simplify both sides of the equation if needed.

For exponential equations like this, once the bases are the same, you can equate the exponents directly, making it a simple linear equation to solve. In our exercise, we equated \(2x\) to \(4\) and then solved for \(x\). Practice identifying these steps and apply them to other similar problems.

Base and Exponent

Understanding base and exponent is crucial in working with exponential equations. The base is the number that is being multiplied, and the exponent tells us how many times the base is used in the multiplication.

• In the expression \(6^{2x}\), \(6\) is the base.
• The \(2x\) is the exponent, indicating that the base \(6\) is raised to the power determined by \(2x\).

When both sides of an equation have the same base, like \(6\) here, comparing the exponents can simplify the solving process greatly. This is because, if \(a^m = a^n\), then \(m = n\), assuming \(a\) is not zero. This principle is what allows us to equate the exponents directly without further manipulation.

Algebraic Manipulation

Algebraic manipulation involves rearranging and transforming equations to isolate the variables you're interested in solving for. In our example, once we equated the exponents \(2x = 4\), we used algebraic manipulation to find \(x\):

• Divide both sides of the equation by the coefficient of \(x\) (which is \(2\)).
• Perform the division to simplify the equation: \(x = \frac{4}{2} = 2\).

This straightforward technique is common in solving linear equations. Remember these steps as you tackle other algebra equations, as they form a foundation for more complex manipulations in mathematics.