Problem 103 - A Graphical Approach to College Algebra

Question

The equations are identities because they are true for all real numbers. Use properties of logarithms to simplify the expression on the left side of the equation so that it equals the expression on the right side, where \(x\) is any real number. $$\ln |x+\sqrt{x^{2}+3}|+\ln |x-\sqrt{x^{2}+3}|=\ln 3$$

Step-by-Step Solution

Answer

Using logarithm properties and simplifying, both sides equal \( \ln 3 \).

Step 1: Understand the Left-Side Expression

We are given the left side of the equation: \( \ln |x+\sqrt{x^{2}+3}|+\ln |x-\sqrt{x^{2}+3}| \). This is a sum of logarithms.

Step 2: Apply the Product Property of Logarithms

The property \( \ln a + \ln b = \ln (a \cdot b) \) allows us to combine the logarithms into a single logarithm: \[ \ln \big(|x+\sqrt{x^{2}+3}| \cdot |x-\sqrt{x^{2}+3}|\big). \]

Step 3: Simplify the Expression Inside the Logarithm

The expression inside the logarithm, \( |x+\sqrt{x^{2}+3}| \cdot |x-\sqrt{x^{2}+3}| \), can be simplified using the difference of squares identity: \( (a+b)(a-b) = a^2 - b^2 \). Here, it's \( x^2 - (\sqrt{x^{2}+3})^2 \).

Step 4: Calculate the Difference of Squares

Calculate the expression \( x^2 - (\sqrt{x^{2}+3})^2 \). This simplifies to \( x^2 - (x^2 + 3) = -3 \). Thus, it becomes \( | -3 | \), which evaluates to \( 3 \) since absolute value makes negative numbers positive.

Step 5: Compare and Confirm with the Right Side

Now we have simplified the left side to \( \ln(3) \). This is exactly the expression given on the right side of the original equation, confirming it's an identity.

Key Concepts

Properties of Logarithms

Logarithmic identities rely on properties of logarithms to simplify expressions. One crucial property is the product property, which states: \( \ln a + \ln b = \ln (a \cdot b) \). This allows us to combine two separate logarithmic terms into one.
By combining the logarithms, it simplifies working with the equation. It's like condensing two separate sentences into one, which can make complex expressions easier to manage.

• Sum to Product: Use this when you have a sum of logarithms; it becomes a product inside the log.
• Product to Single: Simplifies the equation, making it manageable.

Once the equation is simplified into a single logarithm, it's much easier to see what lies within its argument and proceed with further simplification methods. Remember, simplifying using properties like this is often the first step in solving logarithmic equations.

Difference of Squares

The difference of squares is a useful mathematical identity that helps simplify expressions like the one inside our logarithm. The formula is: \( a^2 - b^2 = (a+b)(a-b) \). It states that the difference between two squares can be expressed as a product of a sum and a difference.
In our exercise, the expression \( |x+\sqrt{x^{2}+3}| \cdot |x-\sqrt{x^{2}+3}| \) is simplified using this identity. Here, you identify \( a = x \) and \( b = \sqrt{x^2 + 3} \). The expression becomes:

• \( x^2 - (x^2+3)^2 \): Simplifies to \( -3 \).
• Usefulness: This identity reveals hidden relationships within algebraic expressions.

Recognizing patterns like the difference of squares allows for transforming complex expressions into simpler, more manageable forms. This transformation can then be carried forward to solve or further simplify the problem.

Absolute Value

The absolute value of a number is its distance from zero on the number line. It ensures that whatever is inside the absolute value symbols is always non-negative. This is crucial when simplifying expressions, particularly in our example, where the result of the difference of squares was \( -3 \).
The absolute value of \( -3 \) is \( | -3 | = 3 \), neatly solving the problem with its positive form. Absolute value helps to ensure deals with any negativity, literally.

• Always Positive: Absolute values turn negatives into positives.
• Use in Equations: Helps in managing negative results in expressions, aligning with positive requirements.

By understanding absolute values, it's easier to handle expressions that might otherwise produce troublesome negative results when working through simplifications or solving equations. By recognizing when absolute value is needed, you can ensure that all results fit the necessary criteria of the problem.