Problem 45 - Precalculus

Question

Solve the logarithmic equations exactly. $$\ln x^{2}-\ln 9=0$$

Step-by-Step Solution

Answer

The solutions are \( x = 3 \) and \( x = -3 \).

Step 1: Use Logarithm Properties

Apply the property of logarithms: \( \ln a - \ln b = \ln \left( \frac{a}{b} \right) \). This gives us \( \ln \left( \frac{x^2}{9} \right) = 0 \).

Step 2: Exponentiate to Remove Logarithm

To solve for \( x \), exponentiate both sides of the equation: \( e^{\ln \left(\frac{x^2}{9}\right)} = e^0 \). This simplifies to \( \frac{x^2}{9} = 1 \), because \( e^0 = 1 \).

Step 3: Solve for \( x^2 \)

Multiply both sides by 9 to isolate \( x^2 \): \( x^2 = 9 \).

Step 4: Solve for \( x \)

Take the square root of both sides to solve for \( x \). Remember that both positive and negative roots are valid: \( x = 3 \) or \( x = -3 \).

Key Concepts

Logarithmic Properties

Understanding logarithmic properties is essential when solving logarithmic equations. A fundamental property is that the logarithm of a quotient can be expressed as the difference of the logarithms. Specifically, \( \ln a - \ln b = \ln \left( \frac{a}{b} \right) \). This property allows us to combine or separate logarithms effectively.

In our equation, \( \ln x^2 - \ln 9 = 0 \), we use this property to simplify the left side. By combining the logarithms, we arrive at \( \ln \left( \frac{x^2}{9} \right) = 0 \). This expression is now in a simpler form, which is crucial for further solving steps. Recognizing and applying this property can make seemingly complex equations more manageable.

Exponentiation

Exponentiation is an inverse operation to taking a logarithm. It involves raising a base (usually \(e\) for natural logarithms) to a power to eliminate the logarithmic expression. When you have an equation like \( \ln \left( \frac{x^2}{9} \right) = 0 \), solving for the variable involves removing the logarithm by exponentiating both sides.

In the exercise, we exponentiate to get \( e^{\ln \left( \frac{x^2}{9} \right)} = e^0 \). Since \( e^{\ln(a)} = a \) and \( e^0 = 1 \), the equation simplifies to \( \frac{x^2}{9} = 1 \).

This process of exponentiation helps us transform a logarithmic equation into a simple algebraic equation, allowing for straightforward further manipulation and calculation.

Square Roots

Square roots are used to find values that, when multiplied by themselves, equal a given number. In the final step of solving the equation, the simplified form \( x^2 = 9 \) requires us to find \(x\).

Taking the square root of both sides results in \( x = 3 \) or \( x = -3 \), because both \(3\) and \(-3\) squared will return \(9\). It's important to consider both positive and negative roots when dealing with square roots, especially since they represent different solutions.

This step reinforces the concept that taking a square root yields two potential solutions, reflecting the dual nature of quadratic equations. Understanding this concept ensures accurate and comprehensive solutions in mathematical problem-solving.