Problem 62 - Algebra 2

Question

REVIEW Which equation is equivalent to \(\log _{4} \frac{1}{16}=x ?\) $$ \begin{array}{l}{\mathbf{F} \frac{1^{4}}{16}=x^{4}} \\\ {\mathbf{G}\left(\frac{1}{16}\right)^{4}=x} \\ {\mathbf{H} \quad 4^{x}=\frac{1}{16}} \\ {\mathbf{J} \quad 4^{\frac{1}{16}}=x}\end{array} $$

Step-by-Step Solution

Answer

The equivalent equation is \( 4^{x} = \frac{1}{16} \). (Option H)

Step 1: Understanding Logarithmic Expression

The given logarithmic expression is \( \log_{4} \frac{1}{16} = x \). Here, we need to find the value \( x \) such that \( 4^{x} = \frac{1}{16} \).

Step 2: Identify the Equivalent Exponential Form

Logarithmic expressions can be rewritten as exponential expressions. The expression \( \log_{b}(a) = c \) is equivalent to \( b^{c} = a \). In this case, \( \log_{4}\frac{1}{16} = x \) becomes \( 4^{x} = \frac{1}{16} \).

Step 3: Match the Options

Among the given options, compare them with the exponential form \( 4^{x} = \frac{1}{16} \). Notice that option \( \mathbf{H} \) matches exactly this form: \( 4^{x} = \frac{1}{16} \).

Step 4: Identify the Correct Answer

Since option \( \mathbf{H} \) \( 4^{x} = \frac{1}{16} \) exactly represents the converted exponential form of the logarithmic equation, it is equivalent to \( \log_{4}\frac{1}{16} = x \).

Key Concepts

Equivalent Equations

When working with logarithmic expressions, it's essential to understand the concept of equivalent equations. Two equations are equivalent if they express the same relationship between numbers or variables, even though they might look different at first glance.

In the context of the given problem, we start with the logarithmic equation \( \log_{4} \frac{1}{16} = x \). This can be transformed into an equivalent exponential equation. By converting it into this form, we preserve the relationship between the base, exponent, and result.

Here's the connection:

• The exponential form helps us validate that different equation forms convey the same meaning.
• Identifying equivalent equations is crucial for solving and understanding mathematical problems.

Exponential Form

The exponential form of a number tells us how many times to multiply the base by itself. It's a fundamental way of representing numbers and expressions, especially in logarithmic and exponential problems.

In this exercise, we converted the expression \( \log_{4} \frac{1}{16} = x \) into its exponential form: \( 4^{x} = \frac{1}{16} \). This transformation reveals the underlying power relation in a way that's often easier to manipulate and solve.

Understanding how to switch between logarithmic and exponential forms is a powerful mathematical skill:

• It aids in simplifying expressions and problems.
• Exponential forms are commonly used in growth, decay, and various scientific contexts.
• Knowing the equivalence transforms problem-solving into a more visual and intuitive process.

Logarithms and Exponents

Logarithms and exponents are closely linked mathematical concepts. They are inverse operations, meaning that they undo each other. This inverse relationship is crucial for converting between logarithmic and exponential expressions.

When we say \( \log_{b}(a) = c \), it translates to \( b^{c} = a \) in exponential form. Essentially, a logarithm asks the question: "To what power must we raise this base number to obtain a given number?"

Here's why understanding this connection is helpful:

• It provides clarity in solving equations involving rates, such as compound interest or population growth.
• Logarithms simplify complex multiplications into addition problems, which are easier to compute mentally.
• Exponents describe repeated multiplication, a necessary operation for expressing large numbers compactly.

By mastering logarithms and exponents, you gain a deeper understanding of the mathematical relationships and can solve a wide array of practical and theoretical problems.