Problem 49 - Algebra for College Students

Question

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

Step-by-Step Solution

Answer

The solution is \( x = 4 \).

Step 1: Convert the Logarithmic Equation to Exponential Form

The given equation is \( \log_{x} 2 = \frac{1}{2} \). According to the definition of logarithms, we can convert this into its exponential form. This means \( x^{\frac{1}{2}} = 2 \).

Step 2: Isolate x by Squaring Both Sides

In the equation \( x^{\frac{1}{2}} = 2 \), the exponent \( \frac{1}{2} \) is the same as taking the square root. To eliminate the square root, square both sides of the equation: \( (x^{\frac{1}{2}})^2 = 2^2 \). This simplifies to \( x = 4 \).

Key Concepts

Exponential Form

Transforming a logarithmic equation into its exponential form is integral to solving it. The general form of a logarithmic equation is \( \log_{b}(a) = c \), which can be converted into exponential form as \( b^{c} = a \). This transformation is based on the property of logarithms that relates them directly to exponents.
In the given problem, \( \log_{x} 2 = \frac{1}{2} \), we identify \( b \) as the base \( x \), \( a \) as the number 2, and \( c \) as \( \frac{1}{2} \). Applying this rule, we translate the equation into the exponential form \( x^{\frac{1}{2}} = 2 \).
Here's a simple analogy: consider the equation:

• If \( \log_{10}(100) = 2 \), it means \( 10^{2} = 100 \).

So essentially, converting to exponential form allows us to consider the equation as a power, making it straightforward to solve.

Squaring Both Sides

When you encounter an expression like \( x^{\frac{1}{2}} = 2 \), it's essentially saying that the square root of \( x \) is 2. To solve for \( x \), we need to eliminate the square root.
This is done by squaring both sides of the equation. By squaring, it means you multiply an expression by itself. So, squaring \( x^{\frac{1}{2}} \) results in:

• \( (x^{\frac{1}{2}})^2 = x^{1} = x \).
• Squaring 2 results in \( 2^2 = 4 \).

By applying this method, the equation \( x^{\frac{1}{2}} = 2 \) becomes \( x = 4 \).
This is a powerful tool for simplifying equations where variables are under roots, turning a complex expression into a simpler one where solutions are easily seen.

Solving Equations

Solving equations involves employing different strategies to find the value of unknowns. For the provided logarithmic equation, we first turned it into a friendly exponential form. This way, we can easily manipulate it to find \( x \).
Once we reach the equation \( x = 4 \) in the solution process, it's crucial to verify it satisfies the original equation. Verification serves as a way to ensure accuracy and avoid mistakes.

• Substitute \( x = 4 \) back into the logarithmic form: \( \log_{4} 2 \).
• Check if it equals \( \frac{1}{2} \), affirming the solution is correct.

This logical approach underscores a systematic procedure: converting equations forms, isolating the variable, and confirming results by plugging solutions back. These methods are not just math tricks but fundamental to solving equations reliably across various types of problems.