Problem 73 - Intermediate Algebra

Question

Solve for \(x .\) See Example 3. $$ \log _{x} \frac{\sqrt{3}}{3}=\frac{1}{2} $$

Step-by-Step Solution

Answer

\( x = \frac{1}{3} \).

Step 1: Convert Logarithmic Equation to Exponential Form

The logarithmic equation given is \( \log_{x} \frac{\sqrt{3}}{3} = \frac{1}{2} \). To convert this to an exponential equation, recall that \( \log_{b}(y) = z \) means \( b^{z} = y \). Therefore, \( x^{1/2} = \frac{\sqrt{3}}{3} \).

Step 2: Square Both Sides

To remove the square root on the left, square both sides of the equation: \[ (x^{1/2})^{2} = \left(\frac{\sqrt{3}}{3}\right)^{2} \] This simplifies to \( x = \frac{3}{9} \) since \( \left(\frac{\sqrt{3}}{3}\right)^{2} = \frac{3}{9} \).

Step 3: Simplify the Expression

Now, simplify \( \frac{3}{9} \) to find the solution for \( x \): \[ x = \frac{1}{3} \] Thus, \( x = \frac{1}{3} \).

Key Concepts

Exponential Form

Understanding exponential form is key when working with logarithmic equations. It's the method of expressing repeated multiplication of the same factor. For instance, if you have a base, say, \( b \), raised to an exponent, \( z \), it means \( b \) is multiplied by itself \( z \) times.

In simpler terms, if you're given a logarithmic equation like \( \log_{b}(y) = z \), you can convert it to **exponential form**. This means you rewrite it as \( b^{z} = y \). This transformation helps solve equations by switching from a logarithm to a power.

• **Base (b):** The number that is raised to a power.
• **Exponent (z):** The power to which the base is raised.
• **Result (y):** The outcome of this multiplication.

This connection between logarithms and exponents is foundational in understanding and solving logarithmic equations. Transforming to exponential form makes it easier to manipulate the expressions and move to the next steps in solving the equation.

Square Roots

Square roots are often encountered in mathematical problems and can sometimes complicate operations. A square root is essentially the opposite of squaring a number. For a number \( a \), its square root is expressed as \( \sqrt{a} \), meaning a number that when multiplied by itself yields \( a \).

Relation to Logarithmic Equations

In the context of logarithmic equations, square roots often appear as part of the terms that require adjustment to solve the problem. In our exercise, squaring both sides of the equation \( (x^{1/2})^{2} = (\frac{\sqrt{3}}{3})^{2} \) helped remove the square root on the left side, simplifying the solution process.

• Steps in Square Root Elimination:
• Identify the square root within the equation.
• Square both sides to eliminate the square root.
• Resulting equation becomes easier to work with.

Understanding how to handle square roots in equations is invaluable. It allows for simplification and helps reach the final solution more straightforwardly.

Simplifying Fractions

Simplifying fractions is a fundamental skill in algebra that finds relevance across various types of equations, including logarithmic equations. A fraction is simplified when it is expressed in its lowest terms, which means reducing the numerator and denominator to their smallest possible values while the fraction remains equivalent.

When faced with a fraction \( \frac{a}{b} \), you simplify it by finding the greatest common divisor (GCD) of \( a \) and \( b \), then dividing both by that number.

Application in the Solution

In the problem we solved, simplifying \( \frac{3}{9} \) to \( \frac{1}{3} \) was crucial. Here’s how:

• Identify greatest common divisor (GCD) of 3 and 9, which is 3.
• Divide both the numerator and the denominator by this GCD.
• Resulting fraction is \( \frac{1}{3} \), offering a clearer answer.

Simplifying fractions not only clarifies the mathematical solution but also ensures accuracy in arriving at the correct solution for any given problem.