Problem 278 - Calculus Volume 1

Question

For the following exercises, solve the exponential equation exactly. $$ 8^{x}=4 $$

Step-by-Step Solution

Answer

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

Step 1: Rewrite the equation with the same base

The first step is to express both sides of the equation using the same base. Since both 8 and 4 are powers of 2, we can rewrite 8 as \(2^3\) and 4 as \(2^2\). This gives us the equation: \((2^3)^x = 2^2\).

Step 2: Simplify the expression

Apply the power of a power rule, \((a^m)^n = a^{m\cdot n}\), to simplify the left side: \((2^3)^x = 2^{3x}\). Our equation is now \(2^{3x} = 2^2\).

Step 3: Set the exponents equal

Since the bases are the same, we can set the exponents equal to each other. This gives us the equation \(3x = 2\).

Step 4: Solve for x

To solve for \(x\), divide both sides of the equation by 3: \(x = \frac{2}{3}\).

Key Concepts

Exponentiation

Exponentiation involves raising a number, called the base, to a power, which is the exponent. It simplifies repeated multiplication into a shorthand form.
For example, in the expression \(8^x\), 8 is the base, and \(x\) is the exponent. This means you multiply 8 by itself \(x\) times. Understanding exponentiation is crucial because it lays the foundation for solving exponential equations, where you need to isolate the variable that appears as an exponent.
When given an equation like \(8^x = 4\), we seek to find the value of \(x\) that satisfies this condition. Transforming these values using the same base allows us to manipulate the equation more easily.

Algebraic Manipulation

Algebraic manipulation involves rearranging and rewriting expressions to isolate variables and solve equations. In solving exponential equations, this often means rewriting terms and simplifying them. Take the equation \(8^x = 4\). First, we need to express both sides with a common base. We notice both 8 and 4 are powers of 2. Rewrite 8 as \(2^3\) and 4 as \(2^2\). This converts our equation to \((2^3)^x = 2^2\), making it easier to isolate \(x\).By expressing equations with a common base, algebraic manipulation becomes straightforward. It allows us to focus on the exponents, leading us to the next step in finding the solution. Additionally, using properties like the power of a power rule simplifies complex expressions during manipulation, leading closer to finding the variable's value.

Power of a Power Rule

The power of a power rule is a key concept in exponentiation that helps simplify expressions. This rule states that when raising a power to another power, you multiply the exponents. Mathematically, it is represented as:

• \((a^m)^n = a^{m \cdot n}\)

Applying this rule makes complex expressions more manageable in equations. In our example with \((2^3)^x = 2^2\), the left side simplifies using the power of a power rule to \(2^{3x}\).
Now, the equation is \(2^{3x} = 2^2\). By applying this rule, we align the equation to solve for \(x\). Since the bases are equal, we set their exponents equal too: \(3x = 2\). Solving this gives us \(x = \frac{2}{3}\). Understanding and applying the power of a power rule is essential for simplifying and solving exponential equations efficiently.