Problem 278 - Calculus Volume 1

Question

Solve the exponential equation exactly. \(8^{x}=4\)

Step-by-Step Solution

Answer

The solution is \(x = \frac{2}{3}\).

Step 1: Express Both Sides with the Same Base

First, express both sides of the equation using the same base. Notice that 8 and 4 can be expressed as powers of 2. Specifically, we have: \[ 8 = 2^3 \] \[ 4 = 2^2 \] Therefore, we can rewrite the equation \(8^x = 4\) as:\[ (2^3)^x = 2^2 \]

Step 2: Apply Power of a Power Rule

Using the power of a power rule \((a^m)^n = a^{m imes n}\), simplify the left side:\[ (2^3)^x = 2^{3x} \] This leads to the equation:\[ 2^{3x} = 2^2 \]

Step 3: Equate the Exponents

Since the bases are the same, we can set the exponents equal to each other:\[ 3x = 2 \]This is because if \(a^m = a^n\), then \(m = n\).

Step 4: Solve for x

Solve the equation \(3x = 2\) by dividing both sides by 3:\[ x = \frac{2}{3} \]

Key Concepts

Power of a Power Rule

When dealing with exponential terms, the power of a power rule is incredibly useful. This rule states that when you have a power raised to another power, you multiply the exponents together. For example, if you have

• \((a^m)^n\), it becomes \(a^{m \times n}\).

This is what happens in the original exercise where \((2^3)^x\) becomes \(2^{3x}\). Understanding this rule makes it easier to simplify and solve exponential equations.
It's important to remember to handle the base appropriately. The base remains unchanged while we manipulate the exponents. This trick of multiplying exponents can turn seemingly complex equations into simpler ones. By using the power of a power rule, we break down the problem into more manageable parts.

Exponents

Exponents are a fundamental part of mathematics, representing repeated multiplication. In the equation \(8^x = 4\), the exponents are the key elements that need to be manipulated to solve the equation. Exponents let you express powers in a compact way, such as:

• \(8 = 2^3\)
• \(4 = 2^2\)

Using this expression, the original equation can be rewritten to share the same base.Exponents follow specific rules, such as the power of a power rule, product of powers, and the quotient of powers. By expressing numbers with the same base, you can solve equations by equating just the exponents. Always try to express complex terms in exponential form; it makes solving equations more intuitive and less error-prone.

Solving Equations

Solving exponential equations requires a systematic approach. The goal is to isolate the variable by equating the exponents, which is only possible if the bases are the same. Here’s an easy breakdown of how it’s done:

• First, express all terms with the same base. In our exercise, that was the number 2.
• Apply relevant exponent rules, like the power of a power, to simplify the equation.
• With a consistent base, set the exponents equal to each other. This allows for a straightforward algebraic solution for the variable.

Ultimately, solving for \(x\) involves translating the problem into a simpler mathematical statement. After expressing the equation with the same base, equating the exponents provides a linear equation that's easy to solve, as seen with \(3x = 2\). Isolating \(x\) by dividing by 3 gives the final answer: \(x = \frac{2}{3}\). This method highlights the elegance of mathematical techniques that turn challenging problems into simpler tasks.