Problem 27 - Algebra A Combined Function

Question

By inspection, find the value for \(x\) that makes each statement true. \(2^{x}=8\)

Step-by-Step Solution

Answer

The value of \(x\) is 3.

Step 1: Understand the Problem

We need to determine the value of \(x\) such that \(2^x = 8\). This means finding which power of 2 results in 8.

Step 2: Express 8 as a Power of 2

We know from basic exponentiation that \(2^3 = 8\). Thus, we can express 8 as a power of 2 by writing it as \(2^3\).

Step 3: Set the Exponents Equal

Since \(2^x = 8\) and we have expressed 8 as \(2^3\), we can set the exponents equal: \(x = 3\).

Step 4: Verify the Solution

Substitute \(x = 3\) back into the original equation: \(2^3 = 8\). This confirms that our solution is correct, as both sides of the equation are equal.

Key Concepts

Power of a Number

The concept of 'power of a number' is fundamental in mathematics. It involves raising a base number to an exponent to express repeated multiplication. Here's how it works:

• The 'base' is the number being multiplied, and in our example, the base is 2.
• The 'exponent' tells us how many times to multiply the base by itself. If the exponent is 3, it means \(2 \times 2 \times 2\).
• The result of this multiplication is called the 'power'. For \(2^3\), the power is 8.

Raising a number to a power simplifies writing large products of the same number. Instead of writing out the multiplication, we use exponents to denote the total count of multiplications.

Solving Exponential Equations

Solving exponential equations involves finding the unknown exponent when the base and power are known. Here, the aim is to find the value of \(x\) in the equation \(2^x = 8\).Steps to solve such an equation include:

• Rewrite the power: Express the number on the right side of the equation as a power of the base, if possible. For instance, write 8 as \(2^3\).
• Equal exponents imply equality: Once expressed as the same base, equate the exponents. Here, since \(2^x = 2^3\), we infer \(x = 3\).

This process simplifies finding precise solutions without exhaustive trial and error. It is especially handy for equations with the same base on both sides of the equation.

Basic Exponentiation

Basic exponentiation is the simplest form of exponent arithmetic that deals with positive integer exponents. It provides a foundation for understanding more advanced math concepts.To perform basic exponentiation:

• Identify the base and exponent: Know which number is repeated and how many times. For instance, in \(2^3\), 2 is the base and 3 is the exponent.
• Multiply repeatedly: Multiply the base by itself as many times as the exponent suggests: \(2 \times 2 \times 2 = 8\).
• Understand special cases: Any number raised to the power of zero is 1, and a number raised to the power of one is itself. For example, \(2^0 = 1\) and \(2^1 = 2\).

Learning basic exponentiation is crucial to tackling more complex functions and equations later in mathematics.