Problem 52 - Elementary and Intermediate Algebra

Question

Solve for \(x\). See Example 3 . $$ \log _{x} 9=2 $$

Step-by-Step Solution

Answer

The solution is \( x = 3 \).

Step 1: Understand the Logarithmic Equation

The given equation is \( \log_{x} 9 = 2 \). This specifies that when \( x \) is raised to the power of 2, the result should be 9. In logarithmic terms, this means that \( x^2 = 9 \).

Step 2: Convert the Logarithmic Equation to Exponential Form

According to the properties of logarithms, \( \log_{x} 9 = 2 \) can be rewritten in exponential form as \( x^2 = 9 \). This is because logarithms are the exponents to which the base must be raised to yield a given number.

Step 3: Solve the Exponential Equation

The equation from Step 2 is \( x^2 = 9 \). To solve for \( x \), take the square root of both sides of the equation: \( x = \sqrt{9} \). This results in two possible solutions: \( x = 3 \) or \( x = -3 \).

Step 4: Determine the Valid Solution

Since \( x \) must be a positive number as it is the base of a logarithm (bases of logarithms must be positive), the valid solution is \( x = 3 \).

Key Concepts

Properties of Logarithms

Logarithms are a powerful tool in mathematics; they might seem complex at first, but they are essential for solving equations that involve exponential growth. Understanding the properties of logarithms is crucial for converting logarithmic expressions into exponential equations, making problem-solving much easier.

Some important properties include:

• The Product Rule: This states that the logarithm of a product is equal to the sum of the logarithms of the factors: \( \log_b (MN) = \log_b M + \log_b N \).


• The Quotient Rule: This indicates that the logarithm of a quotient is the difference of the logarithms: \( \log_b \left(\frac{M}{N}\right) = \log_b M - \log_b N \).


• The Power Rule: This asserts that the logarithm of a number raised to an exponent is that exponent times the logarithm of the base: \( \log_b (M^n) = n \cdot \log_b M \).

These properties help us simplify and solve logarithmic equations. In our specific exercise, we recognize that the equation \( \log_x 9 = 2 \) can be transformed using the definition and properties of logarithms to find the base, \( x \), in the exponential form.

Exponential Equations

An exponential equation is one in which variables appear as exponents. Solving such equations often involves switching between logarithmic and exponential forms. Understanding how to make this conversion is key.

In the equation \( x^2 = 9 \), we are dealing with an exponential expression where 9 is the result of base \( x \) raised to the power of 2. Solving this requires removing the exponent by applying the inverse operation, which is taking the square root. This operation gives us potential solutions:

• \( x = 3 \)
• \( x = -3 \)

However, given the context of logarithms, and knowing bases must be positive, we confirm that only \( x = 3 \) is valid. The process of transforming logarithmic form to get an exponential equation is critical in identifying potential solutions that meet the conditions given in a problem.

Solving Equations with Logarithms

Solving equations involving logarithms can often be broken down into more manageable steps. By expressing a logarithmic equation in exponential form, you simplify the problem to one of basic algebra.

In our example, the equation \( \log_x 9 = 2 \) translates to \( x^2 = 9 \) in exponential form by applying the understanding that logarithmically, \( \log_b Y = X \) can be rewritten as \( b^X = Y \). These transformations help us shift from working with logs to familiar arithmetic operations,
such as:

• Converting the logarithmic equation into an equation involving powers


• Using algebraic methods like factoring, or in our example, taking square roots

After identifying potential solutions, it is crucial to consider the properties of logarithmic bases to ensure validity. In this problem, only positive values for \( x \) make sense, as logarithmic bases must be positive numbers. Following these steps will improve solving both simple and complex equations involving logarithms.