Problem 60 - Algebra A Combined Function

Question

Solve. $$ \log _{2 / 3} x=2 $$

Step-by-Step Solution

Answer

The solution is \( x = \frac{4}{9} \).

Step 1: Understanding the Logarithm Equation

The given expression is \( \log_{2/3} x = 2 \). This implies we need to find the value of \( x \) such that when \( 2/3 \) is raised to what power, it results in \( x \), and that power is 2.

Step 2: Transforming the Equation to Exponential Form

Recall that the logarithmic equation \( \log_b a = c \) can be rewritten in exponential form as \( b^c = a \). Applying this to our equation, we have \( \left(\frac{2}{3}\right)^2 = x \).

Step 3: Calculate the Exponential Value

Now we calculate \( \left(\frac{2}{3}\right)^2 \). This means we multiply \( \frac{2}{3} \) by itself: \[ \left(\frac{2}{3}\right) \times \left(\frac{2}{3}\right) = \frac{4}{9} \].

Step 4: Conclude the Solution

The expression evaluates to \( x = \frac{4}{9} \). Therefore, \( \log_{2/3} x = 2 \) translates to \( x = \frac{4}{9} \) in exponential form.

Key Concepts

Exponential Form

Understanding exponential form is key to solving logarithmic equations. The exponential form is a way to express equations with roots. It's highly useful for solving equations like logarithms. Consider the logarithmic equation \( \log_b a = c \). This means that \( b \), the base, raised to the exponent \( c \) is equal to \( a \). In our problem, \( \log_{2/3} x = 2 \), the exponential form is \( \left(\frac{2}{3}\right)^2 = x \). This method gives us a straightforward way to find the value of \( x \). Transforming equations from their logarithmic to exponential forms simplifies them considerably, moving us from a log operation to simple multiplication.

Logarithm Properties

Logarithms come with a set of properties that can simplify complex calculations. These properties help to transform and simplify logarithmic expressions, make calculations easier, and offer deeper insights into relationships between different quantities. Here are some basic properties of logarithms that are often used:

• Product Property: \( \log_b(xy) = \log_b x + \log_b y \)
• Quotient Property: \( \log_b\left(\frac{x}{y}\right) = \log_b x - \log_b y \)
• Power Property: \( \log_b(x^c) = c \log_b x \)

By using these properties, logarithms become much easier to handle. When transformations are necessary, such as changing a logarithmic form to an exponential one, these properties make the process seamless and comprehensible. They are tools for breaking down and reconstructing equations in a simplified form.

Algebra Problem Solving

Tackling algebra problems might appear challenging, but using systematic approaches can make it manageable. Combining algebra with concepts like logarithms requires understanding and applying formulas carefully. For instance, the problem of solving \( \log_{2/3} x = 2 \) involves transitioning from a logarithm to an exponential function. This shift is crucial in simplifying the problem. Here’s a general strategy to solve similar algebra problems:

• **Identify the Equation Form:** Determine whether it's linear, quadratic, logarithmic, etc.
• **Transform as Necessary:** Convert between logarithmic and exponential forms if needed.
• **Simplify:** Use properties of logarithms and exponents to simplify the problem.
• **Solve Sequentially:** Perform step-by-step operations, ensuring each step logically follows the last one.

Solving such equations sharpens problem-solving skills and deepens algebraic understanding. Piece by piece, these systematic solutions pave the way to clearer comprehension and mastery of algebraic concepts.