Problem 27 - Algebra A Combined Function

Question

Solve. $$ \frac{1}{4}=2^{3 x} $$

Step-by-Step Solution

Answer

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

Step 1: Understanding the problem

We need to solve the equation \( \frac{1}{4} = 2^{3x} \). This means we are looking for the value of \( x \) that satisfies the equation.

Step 2: Expressing fractions as powers of 2

Rewrite \( \frac{1}{4} \) as a power of 2. Since \( \frac{1}{4} = 2^{-2} \), we can write the equation as \( 2^{-2} = 2^{3x} \).

Step 3: Using the property of exponents

Because the bases are the same, we can set the exponents equal to each other. Thus, we have \( -2 = 3x \).

Step 4: Solving for x

Isolate \( x \) by dividing both sides of the equation by 3, yielding \( x = \frac{-2}{3} \).

Key Concepts

Understanding Exponents

Exponents are essential in mathematics as they allow us to express repeated multiplication concisely. For instance, instead of writing 2 multiplied by itself several times, we can use the exponent format like \( 2^3 \) to mean \( 2 \times 2 \times 2 \). The small number, located at the top right of the base, is called the exponent or power, and the number being multiplied is the base. In our exercise, the base is 2, and the exponent is \( 3x \). This equation involves finding an unknown exponent that makes two expressions equal. Understanding how to manipulate these exponents is crucial when solving exponential equations. When dealing with exponential equations, a common technique involves rewriting numbers to have the same base, which simplifies comparisons and calculations.

Powers of 2: Simplifying Expressions

Powers of 2 are quite common, especially in problems involving binary systems or exponential growth. Powers of 2 include numbers like 2, 4, 8, 16, etc. Each represents a multiplication of 2 by itself a certain number of times. For example, \( 2^2 = 4 \) and \( 2^3 = 8 \).In the given exercise, we rewrite \( \frac{1}{4} \) as a power of 2. Recognizing that \( \frac{1}{4} \) is equivalent to \( 2^{-2} \) lets us restate the equation as \( 2^{-2} = 2^{3x} \). This step is crucial because it allows us to directly compare the exponents once the bases are identical.Using properties of exponents simplifies many algebraic equations:

• If \( a^m = a^n \), then \( m = n \).
• Any number raised to the power of 0 is 1, i.e., \( a^0 = 1 \).
• The reciprocal of a power is a negative exponent, i.e., \( \frac{1}{a^n} = a^{-n} \).

Algebraic Solutions for Exponential Equations

Solving exponential equations often involves a few key steps, which can make even complex problems manageable. With the bases of the exponential terms equalized, the equation converts into a simple algebraic problem. Following our example, we use the property of exponents: if \( a^m = a^n \), we can equate the exponents. This step simplifies our equation from \( 2^{-2} = 2^{3x} \) to \( -2 = 3x \). Now, solving for \( x \) involves basic algebra. We isolate \( x \) by dividing both sides by 3:\[ x = \frac{-2}{3} \]. This example shows how algebraic techniques like equalizing exponents and isolating variables allow us to solve for unknowns in exponential equations. Such approaches are pivotal when addressing a variety of algebraic challenges.