Problem 23 - Algebra A Combined Function

Question

Solve. $$ 16^{x}=8 $$

Step-by-Step Solution

Answer

\(x = \frac{3}{4}\)

Step 1: Express Both Sides with Common Bases

The equation given is \(16^{x} = 8\). First, we need to express both 16 and 8 as powers of 2. We know that \(16 = 2^4\) and \(8 = 2^3\). So the equation can be rewritten using a common base: \( (2^4)^{x} = 2^3 \).

Step 2: Simplify the Exponentiation

Now simplify the left side of the equation using the power of a power rule \((a^m)^n = a^{m \cdot n}\). This rule gives us \((2^4)^{x} = 2^{4x}\). The equation now becomes \(2^{4x} = 2^3\).

Step 3: Equate the Exponents

Since the bases are the same (both are 2), we can equate the exponents on both sides of the equation. Therefore, we have \(4x = 3\).

Step 4: Solve for x

To solve for \(x\), divide both sides of the equation \(4x = 3\) by 4 to isolate \(x\): \(x = \frac{3}{4}\).

Key Concepts

Common Base

When dealing with expressions raised to a power, it can be very helpful to recognize when they share a common base. In our example, we have the equation \(16^x = 8\). To simplify, both numbers should be expressed as powers of the same base.

Here's how you can do it for this problem:

• Notice that both 16 and 8 can be expressed as powers of 2.
• 16 is the same as \(2^4\) because multiplying 2 by itself four times gives 16.
• Similarly, 8 equals \(2^3\) since multiplying 2 together three times results in 8.
• By expressing both sides of the equation with a common base, things become simpler to solve.

This method of finding a common base makes it straightforward later to compare the exponents directly without any complications in further calculations.

Power of a Power Rule

Once we've expressed both sides of our equation using a common base, we can use the power of a power rule to simplify further. The rule states that \((a^m)^n = a^{m \cdot n}\). This is essential when working with exponents that themselves have an exponent.

For our rewritten equation \((2^4)^x = 2^3\), apply the power of a power rule as follows:

• The expression \((2^4)^x\) can be simplified to \(2^{4x}\).
• This operation clears out the parentheses by multiplying the exponents.

Using this rule reduces all the nested expressions to a single power, making the equation much easier to manage.

Equating Exponents

At this stage, your expression should look somewhat like this: \(2^{4x} = 2^3\). Now we've successfully expressed both sides in terms of a common base (2) and simplified the expressions using the power of a power rule.

Because the bases are the same, you can safely equate the exponents. This means:

• If \(a^m = a^n\), then the relationship \(m = n\) must hold true.

In our problem, this tells us that \(4x = 3\).

Equating exponents is a logical leap that is only valid when both sides of the equation have the same base. Once the equation is reduced to comparing the exponents, it's just a matter of simple algebra to solve, as we isolate \(x\) to find that \(x = \frac{3}{4}\). This step is often the most straightforward in these kinds of problems, once you've set up the equation correctly.