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The exponential form is a fundamental way to express equations involving powers. When discussing logarithmic equations, converting to exponential form often simplifies understanding and solving the problem. In an exponential equation, a base is raised to a certain exponent or power. This is expressed as \( b^y = x \), meaning the base \( b \) raised to the power of \( y \) results in the value \( x \).

• The base: In the given problem, our base is \( \frac{2}{3} \).
• The exponent: The exponent here is 2, as represented in the equation \( \log_{2/3} x = 2 \).
• The result: When the base \( \frac{2}{3} \) is squared, we find the value of \( x \).

To convert the logarithmic form into exponential form, take the base of the logarithm and raise it to the power of the result of the logarithm equation. So, the logarithmic equation \( \log_{2/3} x = 2 \) becomes \( x = (2/3)^2 \) in exponential form. This transformation allows us to express the relationship neatly and find \( x \) straightforwardly.

Logarithms are powerful mathematical tools used to solve equations involving exponents. Essentially, a logarithm answers the question: "To what power must the base be raised to yield a specific number?"

• Base: The foundational number that is raised to a power. Here, \( \frac{2}{3} \) is the base.
• Logarithm expression: The notation \( \log_{b}x \) represents the power to which the base \( b \) must be raised to yield the number \( x \).
• Result of log: In our example, the logarithm \( \log_{2/3} x = 2 \) implies that raising \( \frac{2}{3} \) to this power results in \( x \).

Understanding the concept of logarithms helps us navigate problems involving exponential growth or decay, making them a critical concept in both mathematics and real-world applications. Here, by recognizing that \( \log_{2/3} x = 2 \), it indicates that \( x \) is the outcome when \( \frac{2}{3} \) is highly raised to a square (or second power).
This solidifies our grasp on how to manage equations involving logarithmic terms.

Solving equations involves finding a value of the variable that makes the equation true. For logarithmic equations like the one we have here, the solution can often be facilitated by converting to a different form.
Let's break down the process:

• Understanding the Problem: The given equation is \( \log_{2/3} x = 2 \). Our task is to find \( x \).
• Conversion: Convert the equation to exponential form, which helps in visualizing and solving: \( x = (2/3)^2 \).
• Calculation: Compute the exponent: \((2/3)^2 = 4/9 \). This is done by squaring both the numerator and the denominator separately.
• Solution Validation: Check if \( 4/9 \) satisfies the original logarithmic equation \( \log_{2/3} (4/9) = 2 \). Since it does, \( x = 4/9 \) is correct.

Steps like these highlight the logical sequence to follow when tackling logarithmic equations. By understanding each part of the process, students can apply similar techniques to a wide variety of problems involving logarithms and exponents. The key lies in transforming the equation into a form that is easier to solve, calculating accurately, and verifying the results.