Problem 63 - Algebra A Combined Function

Question

Solve. $$ \log _{2} 2^{4}=x $$

Step-by-Step Solution

Answer

The value of \( x \) is 4.

Step 1: Recognize the Logarithm Property

The given exercise involves a logarithm of an exponential expression: \( \log_{2}{2^4} = x \). A fundamental property of logarithms is \( \log_b(b^y) = y \). Here, we will apply this property to simplify the expression.

Step 2: Apply the Logarithm Property

Using the logarithm property \( \log_b(b^y) = y \), we can simplify the expression. Since the base of the logarithm and the base of the power are the same (i.e., both are 2), the expression simplifies directly to \( x = 4 \).

Step 3: Verify the Solution

To verify, understand that \( \log_{2}{2^4} \) asks the question, 'To what power must 2 be raised to obtain \( 2^4 \)?' Clearly, this power is 4, confirming that \( x = 4 \) is correct.

Key Concepts

Exponential Expressions

Exponential expressions are mathematical phrases involving exponents. An exponent tells us how many times a number, known as the base, is multiplied by itself. For example, in the expression \(2^4\), 2 is the base, and 4 is the exponent. This indicates that 2 is multiplied by itself 4 times: \(2 \times 2 \times 2 \times 2\). Understanding exponential expressions is crucial, as they often appear in equations that involve growth, decay, and other real-world scenarios. By mastering the concepts behind exponents, solving complex equations and understanding exponential growth or decay becomes much easier. Exponents also have specific rules that simplify computations. For instance:

• Any number to the power of 0 is 1: \(a^0 = 1\).
• Multiplying with the same base: \(a^m \times a^n = a^{m+n}\).
• Dividing with the same base: \(a^m \div a^n = a^{m-n}\).

Logarithm Properties

Logarithms are the opposites of exponential functions. They help us solve equations where the unknown is an exponent. A logarithm \(\log_b(x)\) answers the question: 'To what power must the base \(b\) be raised, to yield \(x\)?'Logarithm properties are powerful tools that help simplify and solve equations involving exponents. The property used in the example \( \log_{2}{2^4} = x \) is that for any base \(b\), \( \log_b(b^y) = y \). This means if the base of the logarithm matches the base of the exponential expression, the solution is simply the exponent. Other useful properties include:

• Product Rule: \( \log_b(mn) = \log_b(m) + \log_b(n) \)
• Quotient Rule: \( \log_b\left(\frac{m}{n}\right) = \log_b(m) - \log_b(n) \)
• Power Rule: \( \log_b(m^n) = n\log_b(m) \)

Understanding these properties allows you to manipulate and solve logarithmic equations with confidence.

Mathematical Problem Solving

Problem solving in mathematics is a skill developed over time through practice and understanding of core concepts, like exponents and logarithms. When faced with a problem such as \( \log_{2}{2^4} = x \), begin by identifying the type of problem and categorize it—here, it's a logarithmic problem involving an exponential expression. The next step is to apply the properties and rules you've learned. Recognizing familiar patterns, like matching the base of the logarithm to the exponential base, can simplify the problem immediately. Verification is another key problem-solving step. Once a solution is reached, pause to check the accuracy.To verify \(x = 4\) for the example, ask what power of 2 results in \(2^4\). Since the expression directly answers that \(2^4\) is achieved with a power of 4, the solution is confirmed. Effective mathematical problem solving involves:

• Identifying the type of problem and applicable concepts.
• Strategically applying relevant properties or rules.
• Always verifying the result to ensure correctness.

With practice, these steps allow for efficient and confident handling of mathematical challenges.