Problem 50 - Algebra for College Students

Question

Solve each equation. \(\log _{x} 3=\frac{1}{2}\)

Step-by-Step Solution

Answer

The solution is \( x = 9 \).

Step 1: Understand the logarithmic equation

We have the equation \( \log_{x} 3 = \frac{1}{2} \). This means that when \( x \) is raised to the power of \( \frac{1}{2} \), the result is 3.

Step 2: Convert the log equation to an exponential form

Use the definition of a logarithm: if \( \log_{b} a = c \), then \( b^c = a \). Apply this to the equation \( \log_{x} 3 = \frac{1}{2} \), which gives \( x^{\frac{1}{2}} = 3 \).

Step 3: Solve for \(x\)

To get rid of the exponent \( \frac{1}{2} \) or to remove the square root, square both sides of the equation: \( (x^{\frac{1}{2}})^2 = 3^2 \). This simplifies to \( x = 9 \).

Key Concepts

Exponential Form

Understanding the concept of exponential form is essential when dealing with logarithmic equations. The exponential form is another way to express the relationship given by a logarithm. If we have a logarithmic equation like \( \log_{b} a = c \), it means that the base \( b \) raised to the power of \( c \) equals \( a \). For example, in our exercise, \( \log_{x} 3 = \frac{1}{2} \) can be rewritten in exponential form as \( x^{\frac{1}{2}} = 3 \). This reinterpretation makes it easier to solve the equation.

We convert logarithmic equations to exponential form since exponentials can be more intuitive to solve. By rewriting the equation, we can clearly understand the exponential relationship between the terms. This step bridges the gap from recognizing the logarithm to solving the actual equation.

Properties of Logarithms

The properties of logarithms provide us with the tools to manipulate and simplify logarithmic expressions. Some key properties such as the product, quotient, and power rules allow us to work efficiently with complex equations.

• 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: \( \log_{b}(M^n) = n \cdot \log_{b}M \)

While our original exercise focuses on converting and solving the logarithmic equation by another method, these properties are useful for simplifying expressions or solving equations involving multiple terms. Understanding these can also clarify how logarithmic equations may have multiple paths to a solution.

Solving Equations

Solving logarithmic equations often requires converting to exponential form, as in our exercise. Once transformed, equations become more straightforward to solve.

After converting \( \log_{x} 3 = \frac{1}{2} \) to \( x^{\frac{1}{2}} = 3 \), we solve for \( x \) by eliminating the exponent. In this instance, the exponent is \( \frac{1}{2} \), indicative of a square root. To "undo" the square root, we square both sides, leading to \( (x^{\frac{1}{2}})^2 = 9 \). Thus, \( x = 9 \).

Every step in solving requires applying fundamental algebra and understanding of exponentials. By carefully undoing logarithmic and exponential operations, such equations become approachable, allowing for systematic solutions.