Problem 34 - Algebra 2 and Trigonometry

Question

Solve each equation and check. \(5^{x-1}=(0.04)^{2 x}\)

Step-by-Step Solution

Answer

The solution is \(x = \frac{1}{5}\).

Step 1: Recognize the Base Change

The equation given is \(5^{x-1} = (0.04)^{2x}\). First, notice that \(5\) is the base of the first exponent and \(0.04\) is the base of the second exponent. Recognize that \(0.04\) can be rewritten as a power of 5: \(0.04 = \left(\frac{1}{25}\right)\), which can further be written as \( (5^{-2}) \). Now the equation becomes \(5^{x-1} = (5^{-2})^{2x}\).

Step 2: Rewrite Using the Same Base

With both sides of the equation now having a base of 5, we can rewrite it as \(5^{x-1} = 5^{-4x}\). Since the bases are the same, we can set the exponents equal to each other.

Step 3: Set Exponents Equal

Equate the exponents from each side of the equation: \(x-1 = -4x\). Solve for \(x\) by combining like terms and simplifying the equation.

Step 4: Solve for x

Solving \(x-1 = -4x\) leads to the following simplification: First add \(4x\) to both sides to get \(x + 4x - 1 = 0\), which simplifies to \(5x - 1 = 0\). Add 1 to both sides, yielding \(5x = 1\). Finally, divide both sides by 5, resulting in \(x = \frac{1}{5}\).

Step 5: Check the Solution

Substitute \(x = \frac{1}{5}\) back into the original equation to verify: First calculate the left side \(5^{x-1}\): \(5^{\frac{1}{5}-1} = 5^{-rac{4}{5}}\). Now calculate the right side \((0.04)^{2x}\): \((0.04)^{2 \times \frac{1}{5}} = (0.04)^{\frac{2}{5}}\). Since \(0.04 = \left(\frac{1}{25}\right)= (5^{-2})\), this results in \((5^{-2})^{\frac{2}{5}} = 5^{-(\frac{4}{5})}\). Both sides evaluate to \(5^{-\frac{4}{5}}\), confirming our solution \(x = \frac{1}{5}\) is correct.

Key Concepts

Base Rewriting

Understanding how to rewrite bases is essential for solving exponential equations effectively. In the equation \(5^{x-1} = (0.04)^{2x}\), the challenge starts with the mismatched bases: 5 and 0.04. The skill here is recognizing equivalent bases. For example:

• 0.04 can be rewritten as a fraction: \(\frac{1}{25}\).
• This fraction can then be expressed as a power of 5: \(5^{-2}\).

By converting 0.04 into \(5^{-2}\), you simplify the problem to work with just one base, which in this case is 5. Rewriting the base helps set the stage for further simplification.

Exponent Rules

Once the bases are rewritten to be the same, exponent rules come into play. The main rules involve multiplying exponents and understanding negative exponents. Here’s how they work:

• When you raise a power to another power, you multiply the exponents: \((5^{-2})^{2x} = 5^{-4x}\).
• If you ever encounter a negative exponent like \(5^{-4x}\), it signifies division: \(5^{-4x} = \frac{1}{5^{4x}}\).

Understanding these rules aids in simplifying complex expressions and solving equations successfully.

Simplifying Exponents

Simplifying exponents involves handling expressions that might look complicated at first. Consider the equation: \(5^{x-1} = 5^{-4x}\). This is simplified by:

• Setting the exponents equal since the bases are the same.
• Equating \(x-1 = -4x\) allows the elimination of the exponents by focusing directly on solving for \(x\).

This simplification reduces the equation to a linear form, making it easier to find solutions.

Equation Solving Steps

The process of solving equations involves clear, methodical steps. Here is a breakdown of the steps used to solve \(x-1 = -4x\):

• Add \(4x\) to both sides to eliminate the \(-4x\) on the right, resulting in \(5x - 1 = 0\).
• Next, add 1 to both sides to isolate terms with \(x\) on one side: \(5x = 1\).
• Finally, divide both sides by 5 to find \(x\): \(x = \frac{1}{5}\).

Following these steps carefully ensures you achieve the correct solution and fully verify it by substituting back into the original equation to check accuracy.