Problem 104 - A Graphical Approach to Precalculus with Limits

Question

The following equations are identities because they are true for all real mumbers. 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 neal number. $$\ln \left|x^{2}-\sqrt{x^{4}+1}\right|+\ln \left|x^{2}+\sqrt{x^{4}+1}\right|=0$$

Step-by-Step Solution

Answer

The identity holds because both sides simplify to 0.

Step 1: Apply Logarithm Property

The problem involves the sum of two logarithms. For logarithmic expressions, we use the property: \( \ln(a) + \ln(b) = \ln(a \times b) \). Apply this property:\[\ln \left| x^{2} - \sqrt{x^{4}+1} \right| + \ln \left| x^{2} + \sqrt{x^{4}+1} \right| = \ln \left( \left| x^{2} - \sqrt{x^{4}+1} \right| \times \left| x^{2} + \sqrt{x^{4}+1} \right| \right)\]

Step 2: Simplify the Expression Inside the Logarithm

Recognize the expression inside as a difference of squares: \(\left( x^{2} - \sqrt{x^{4}+1} \right) \times \left( x^{2} + \sqrt{x^{4}+1} \right) = \left(x^2\right)^2 - \left(\sqrt{x^{4}+1}\right)^2\)Applying the difference of squares:\[(x^2)^2 - (\sqrt{x^{4}+1})^2 = x^4 - (x^{4} + 1)\]This simplifies to:\[x^4 - x^4 - 1 = -1\]Substitute this back:\[\ln \left| -1 \right| = \ln(1)\]

Step 3: Evaluate the Simplified Logarithmic Expression

\(\ln(1)\) is a well-known value:\[\ln(1) = 0\]Replace the left side of the original equation with this result:\[0 = 0\]This confirms the identity is true as the left side simplifies to the right side.

Key Concepts

Properties of Logarithms

Logarithms have certain properties that make simplifying complex expressions more manageable. One of the most important is the property of adding logarithms, which is very similar to multiplying numbers.
For instance, if you have two terms being added, like \( \ln(a) + \ln(b) \), you can use a property to combine them into a single logarithm: \( \ln(a \times b) \). This simplifies the expression significantly.

• This rule is derived from the inverse nature of logarithms and exponents, where multiplication inside the logarithm corresponds to addition outside.
• Understanding this property helps in tasks like factoring, expanding, and reducing logarithmic equations.
• It's particularly useful when faced with large numbers or when a computation needs simplification.

These properties allow us to recognize patterns and transform expressions efficiently, leading us toward solutions quicker.

Difference of Squares

The difference of squares is another valuable mathematical tool used to simplify expressions. This technique is apparent when an expression takes on the form \(a^2 - b^2\).
Here's how it works: A difference of squares can be factored into the product of a sum and difference, \((a - b)(a + b)\). Recognizing this form enables quick simplification.

• For example, \(x^4 - (x^4 + 1)\) in the previous step simplifies well because it's written in the form of a difference of squares.
• By applying this rule, the expression simplifies to \(-1\), as the terms cancel each other out.

This powerful property helps in reducing complex algebraic equations, allowing direct solutions without extensive calculations. Knowing how to identify and use the difference of squares can save much effort in algebra.

Simplifying Logarithmic Expressions

Simplifying logarithmic expressions involves using both properties of logarithms and factoring techniques like the difference of squares.
Let's break down how these methods work together:
1. Start with identifying components of the expression, such as sums of logarithms or potential factorizations.
2. Apply properties of logarithms, like converting a sum \( \ln(a) + \ln(b) \) into \( \ln(a \times b) \).
3. Use algebraic techniques, such as recognizing a difference of squares to simplify the expression inside the logarithm.

• In our exercise, simplifying the product of terms inside logarithms leads you to an expression that can further reduce down to a known value.
• Here, it matched \( \ln(1) \), which simplifies to '0' directly.

Ultimately, simplifying these expressions involves recognizing patterns and using properties strategically for efficient resolution.
Understanding these processes eases the path to solving many logarithmic equations and other advanced algebra problems, allowing students to grasp complex mathematics confidently and effectively.