Problem 75 - Intermediate Algebra

Question

Find each value of \(x .\) $$ \log _{x} \frac{9}{4}=2 $$

Step-by-Step Solution

Answer

The value of \(x\) is \(\frac{3}{2}\).

Step 1: Understand the Logarithmic Equation

The given equation is \( \log_{x} \frac{9}{4} = 2 \). This means that \( x^2 = \frac{9}{4} \), because a logarithm is the power to which the base must be raised to obtain a given number.

Step 2: Convert Logarithmic Form to Exponential Form

Converting from the logarithmic form to exponential form, we have \( x^2 = \frac{9}{4} \). This tells us what the base \( x \) raised to the power of 2 equals.

Step 3: Solve the Exponential Equation

To find \( x \), we need to solve the equation \( x^2 = \frac{9}{4} \). This involves taking the square root of both sides, which gives us two potential solutions: \( x = \pm \frac{3}{2} \).

Step 4: Consider the Domain of the Logarithm

Remember that the base of a logarithm must be positive and cannot be 1. Hence, we discard \( x = -\frac{3}{2} \), leaving \( x = \frac{3}{2} \) as our valid solution.

Key Concepts

Logarithmic Form to Exponential Form

When working with logarithmic equations, converting them into exponential form is often a crucial step. The given problem started with the equation \( \log_{x} \frac{9}{4} = 2 \). To understand this equation, recall that a logarithm is essentially the inverse of an exponent.
The logarithm tells you the power to which you have to raise the base to get a specific number. In this problem, the base is \( x \), the power is 2, and the result is \( \frac{9}{4} \). Thus, in exponential form, this equation becomes \( x^2 = \frac{9}{4} \).
Converting to exponential form helps simplify the solving process by expressing the equation in terms of an exponent, making it more straightforward to isolate the desired variable.

Exponential Equations

Once the equation is converted into exponential form as \( x^2 = \frac{9}{4} \), it becomes an exponential equation. Exponential equations involve variables in the exponent, but in this case, it simplifies to a more familiar format, a quadratic equation.
To solve \( x^2 = \frac{9}{4} \), we need to take the square root of both sides:

• \( x = \frac{3}{2} \)
• \( x = -\frac{3}{2} \)

While we have two potential solutions, exponential equations often lead to cases that may not always include all solutions, especially in the context of a logarithm as we'll see next.

Domain of Logarithms

The domain of a logarithm refers to the possible values that can be used as its base. Recall the base of a logarithm \( \log_{b} a \) must satisfy certain conditions:

• It must be positive: \( b > 0 \)
• It cannot be equal to 1: \( b eq 1 \)

In the original problem, after solving the exponential equation, we had two solutions: \( x = \frac{3}{2} \) and \( x = -\frac{3}{2} \).
However, due to the domain constraints of the base of a logarithm, \( x = -\frac{3}{2} \) cannot be used as it violates the condition of being positive.
Thus, the only valid solution for this problem, considering the domain of logarithms, is \( x = \frac{3}{2} \). This highlights the importance of always verifying solutions to ensure they conform to the mathematical rules governing logarithmic functions.