Problem 48 - A Graphical Approach to Precalculus with Limits

Question

Solve each equation. Do not use a calculator. $$32^{x}=16^{1-x}$$

Step-by-Step Solution

Answer

The solution is \(x = \frac{4}{9}\).

Step 1: Express Numbers as Powers of a Common Base

The numbers 32 and 16 can both be expressed as powers of 2. Notice that:\[ 32 = 2^5 \]\[ 16 = 2^4 \]

Step 2: Rewrite the Equation with Powers of 2

Substitute the expressions from Step 1 into the original equation:\[ (2^5)^x = (2^4)^{1-x} \]

Step 3: Apply the Power of a Power Rule

Use the exponentiation rule \((a^m)^n = a^{m }n\) to simplify both sides of the equation:\[ 2^{5x} = 2^{4(1-x)} \]

Step 4: Set the Exponents Equal

Since the bases are the same, the exponents must be equal. Set the exponents from both sides of the equation equal to each other:\[ 5x = 4(1-x) \]

Step 5: Distribute and Simplify

Distribute the 4 on the right side of the equation:\[ 5x = 4 - 4x \]Add \(4x\) to both sides to get:\[ 9x = 4 \]

Step 6: Solve for x

Divide both sides by 9 to solve for \(x\):\[ x = \frac{4}{9} \]

Key Concepts

Powers of a Common Base

To effectively solve equations involving exponents, it's crucial to understand the concept of powers of a common base. This approach involves rewriting numbers as powers of the same base, typically using prime factors to find a simple exponent representation.

In our exercise, we successfully express 32 and 16 as powers of 2:

• \(32 = 2^5\)
• \(16 = 2^4\)

By using a common base, we transform complicated exponential equations into more straightforward algebraic expressions, making them much easier to handle without a calculator.

Solving Exponential Equations

Once we have expressed numbers as powers of a common base, the next step is solving the exponential equation. With both sides having the same base, we can equate the exponents.

Consider our rewritten equation:

• \((2^5)^x = (2^4)^{1-x}\)

Simplifying this using exponent rules, we get:

• \(2^{5x} = 2^{4(1-x)}\)

Since the bases are the same, we can equate the exponents:

• \(5x = 4(1-x)\)

This allows us to solve for the unknown variable, here represented as \(x\), by using simple algebraic manipulations such as distributing, adding, and dividing terms.

Power of a Power Rule

An essential tool for working with exponential expressions is the power of a power rule. This rule states that when you raise a power to another power, you multiply the exponents: \((a^m)^n = a^{m \cdot n}\).

In our exercise, applying this rule is especially useful when rewriting the equation:

• \((2^5)^x\) becomes \(2^{5x}\)
• \((2^4)^{1-x}\) becomes \(2^{4(1-x)}\)

By simplifying the equation using the power of a power rule, we bring the equation to a form where the exponents are clear and easily comparable. This method simplifies the process considerably, enabling us to solve for the unknown in a concise manner.