Problem 92 - College Algebra with Modeling and Visualization

Question

Show that $$ \log _{2}(x+\sqrt{x^{2}-4})+\log _{2}(x-\sqrt{x^{2}-4})=2 $$ is an identity. What is the domain of the expression on the left side of the equation?

Step-by-Step Solution

Answer

The identity holds. Domain: \(x \leq -2\) or \(x \geq 2\).

Step 1: Apply Logarithm Properties

We begin by using the logarithm property that allows us to combine two logs with the same base: \( \log_a b + \log_a c = \log_a (bc) \). Thus, we can rewrite the left-hand side of our equation as:\[\log_{2} ((x + \sqrt{x^2 - 4})(x - \sqrt{x^2 - 4})) = 2.\]

Step 2: Simplify the Expression

Now, we need to simplify the expression \((x + \sqrt{x^2 - 4})(x - \sqrt{x^2 - 4})\). This resembles the difference of squares, \((a + b)(a - b) = a^2 - b^2\). Thus:\[(x + \sqrt{x^2 - 4})(x - \sqrt{x^2 - 4}) = x^2 - (x^2 - 4) = 4.\]

Step 3: Evaluate the Logarithm

Substitute the simplified expression back into the logarithm:\[\log_{2}(4) = 2.\]Since \(\log_{2}(4) = \log_{2}(2^2) = 2\), this confirms the equation is an identity.

Step 4: Determine the Domain

The original expression involves square roots, so we need to ensure that the expression under the square root is non-negative. Thus,\[x^2 - 4 \geq 0.\]Solving \(x^2 - 4 = 0\) gives \(x = \pm 2\). The inequality \(x^2 - 4 \geq 0\) is satisfied for \(x \leq -2\) or \(x \geq 2\). These are the domain values that keep the original expression real.

Key Concepts

Logarithm Properties

Logarithm properties are essential tools in simplifying equations involving logarithms. The key property used in the given problem is that which allows the sum of two logarithms with the same base to be expressed as a single logarithm. It states that \( \log_a b + \log_a c = \log_a (b \cdot c) \). This property is derived from the definition of logarithms and the fundamental operations on exponents.

In the exercise, this property helps to condense two separate logarithmic terms on the left-hand side of the equation into one. Specifically, \( \log_{2}(x+\sqrt{x^{2}-4}) + \log_{2}(x-\sqrt{x^{2}-4}) \) becomes \( \log_{2}((x+\sqrt{x^{2}-4})(x-\sqrt{x^{2}-4})) \). This transformation simplifies the evaluation process by reducing the expression to a single logarithm of a simpler mathematical expression.

Domain of a Function

The domain of a function refers to all the possible input values (typically represented as \( x \)) that will produce a real output value. In the context of the given problem, determining the domain involves analyzing the expression inside the logarithms and the square roots. Logarithm functions are defined only for positive real numbers, and square roots are defined for non-negative numbers.

Therefore, for the expression \( \sqrt{x^2 - 4} \) to be defined, the value under the square root, \( x^2 - 4 \), must be greater than or equal to zero. Solving \( x^2 - 4 \geq 0 \), we find the results \( x \leq -2 \) or \( x \geq 2 \), which determines the domain of the entire expression. These intervals ensure that both the square root and the logarithmic expressions are real and valid inputs.

Difference of Squares

The difference of squares is a powerful algebraic tool used to simplify expressions. The formula \( (a + b)(a - b) = a^2 - b^2 \) is employed when two binomial expressions appear in the form of a sum and a difference. In the problem, the expression \((x + \sqrt{x^2 - 4})(x - \sqrt{x^2 - 4})\) matches this pattern.

Applying the difference of squares yields: \[ (x + \sqrt{x^2 - 4})(x - \sqrt{x^2 - 4}) = x^2 - (x^2 - 4) \]. This simplification leads to \( 4 \), transforming the logarithmic equation into a straightforward form \( \log_{2}(4) = 2 \). Through this algebraic step, the completion of the problem is achieved, confirming that the expression resolves to an identity as required.