Problem 4 - College Algebra and Calculus: An Applied Approach

Question

Solve for \(x\).\(4^{x}=\frac{1}{256}\)

Step-by-Step Solution

Answer

The solution to the exponential equation \(4^x = 1/256\) is \(x = -4\).

Step 1: Express \(1/256\) as a Power of 4

Observe that \(1/256 = 4^{-4}\). Both 4 and 256 are powers of 2, which enables us to check this by calculating \(4^{-4} = (2^2)^{-4} = 2^{-8} = 1/256\). Thus the original equation \(4^x = 1/256\) can be rewritten as \(4^x = 4^{-4}\).

Step 2: Compare the Exponents

Now that both sides of the equation are expressed with the same base, we can set the exponents equal to each other: \(x = -4\).

Step 3: Check the Solution

To validate this solution, we can insert \(x = -4\) into the original equation and check if both sides are equal: \(4^{-4} = 1/256\), which is true. Thus, the solution is valid.

Key Concepts

Powers of Numbers

When dealing with powers of numbers, we use an exponent to indicate how many times a number, called the base, is multiplied by itself. For instance, in the expression \(4^x\), the number 4 is the base and \(x\) is the exponent. The exponent tells us the number of times to use the base in a multiplication. Here's a simple breakdown:

• If \(x = 2\), then \(4^x = 4^2 = 4 \times 4 = 16\).
• If \(x = 3\), then \(4^x = 4^3 = 4 \times 4 \times 4 = 64\).

It's important to remember the pattern: as the exponent increases, the result (also known as the power) gets rapidly larger, showcasing exponential growth.

Negative Exponents

In cases like \(4^{-4}\), negative exponents can initially be confusing, but they actually provide a clear mathematical purpose. A negative exponent indicates the reciprocal of the base raised to the absolute value of the exponent. That transforms \(4^{-4}\) into \(\frac{1}{4^4}\). Unlike positive exponents that denote repeated multiplication, negative exponents tell us to divide:

• \(4^1 = \frac{1}{4^{-1}}\) or \(4\)
• \(4^2 = \frac{1}{4^{-2}}\) or \(16\)
• Finally, using \(4^{-4}\), we get \(\frac{1}{256}\)

Negative exponents are a useful tool that helps in simplifying complex expressions and solving exponential equations effortlessly.

Equivalent Expressions

Understanding equivalent expressions is crucial for solving equations like \(4^x = \frac{1}{256}\). Equivalent expressions refer to different expressions that have the same value for all values of the variable. In our problem, expressing \(\frac{1}{256}\) as \(4^{-4}\) allows direct comparison with \(4^x\).Why is this useful?

• Allows simplification of equations: By having identical bases, we can equate the exponents, reducing a complex problem to a simpler form.
• Improves accuracy: Working with one base reduces complexity and eliminates chances for error.

It's all about transforming the equation into a form that is easier to analyze, ultimately leading us to find that \(x = -4\) is the correct and equivalent solution.