Problem 104 - 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. $$\ln \left|x^{2}-\sqrt{x^{4}+1}\right|+\ln \left|x^{2}+\sqrt{x^{4}+1}\right|=0$$

Step-by-Step Solution

Answer

The expression simplifies to 0, matching the right side of the equation.

Step 1: Analyze the Logarithm Property

Recall the property of logarithms that states \( \ln(a) + \ln(b) = \ln(ab) \). Use this property to combine the two logarithmic expressions on the left side of the equation: \( \ln \left| x^2 - \sqrt{x^4 + 1} \right| + \ln \left| x^2 + \sqrt{x^4 + 1} \right| = \ln \left| \left( x^2 - \sqrt{x^4 + 1} \right) ( x^2 + \sqrt{x^4 + 1} \right) \right| \).

Step 2: Apply Difference of Squares

The expression inside the logarithm simplifies using the difference of squares formula: \( (a - b)(a + b) = a^2 - b^2 \). Here, let \( a = x^2 \) and \( b = \sqrt{x^4 + 1} \). Thus, we simplify to \( \ln \left| (x^2)^2 - (\sqrt{x^4 + 1})^2 \right| \).

Step 3: Simplify the Expression

Calculate the difference: \( (x^2)^2 = x^4 \) and \( (\sqrt{x^4 + 1})^2 = x^4 + 1 \). Thus, the expression becomes \( x^4 - (x^4 + 1) = x^4 - x^4 - 1 = -1 \).

Step 4: Evaluate the Logarithm

The expression simplifies to \( \ln| -1 | = \ln(1) \), because the absolute value of \(-1\) is 1.

Step 5: Finalize the Simplification

Since \( \ln(1) = 0 \), we have simplified the left side of the equation to a numerical value of 0, which matches the right side of the equation.

Key Concepts

Properties of Logarithms

Understanding the properties of logarithms is crucial when solving equations that involve them. One of the fundamental properties is that the sum of the logarithms can be combined into a single logarithm of a product. This is written as \( \ln(a) + \ln(b) = \ln(ab) \).
This property allows us to merge multiple logarithmic terms into a single compact expression, making complicated equations easier to manage.
For example, if you have an equation like the one in our exercise involving \( \ln \left|x^{2}-\sqrt{x^{4}+1}\right|+\ln \left|x^{2}+\sqrt{x^{4}+1}\right| \), we can use the rule to simplify it to:

• \( \ln \left| (x^2 - \sqrt{x^4 + 1})(x^2 + \sqrt{x^4 + 1}) \right| \)

By understanding these properties, one can streamline the solving process, leading directly to a simplified result.

Difference of Squares

The difference of squares is a particular algebraic identity that helps in simplifying expressions. It's defined by the formula \((a - b)(a + b) = a^2 - b^2\).
This pattern is noticeable in many mathematical expressions and plays a key role in our exercise.
In this context, we have the terms \(x^2 - \sqrt{x^4 + 1}\) and \(x^2 + \sqrt{x^4 + 1}\), which can be seen as \(a - b\) and \(a + b\). Applying the difference of squares formula, we set:

• \( a = x^2 \)
• \( b = \sqrt{x^4 + 1} \)

Thus, \((x^2)^2 - (\sqrt{x^4 + 1})^2\) simplifies to \(x^4 - (x^4 + 1)\). This simplification is crucial as it reduces complexity and makes the problem much more manageable.

Simplifying Logarithmic Expressions

Simplifying logarithmic expressions involves using algebraic identities and properties to reduce them to their simplest forms. In our exercise, we applied the property of combining logarithms into a single logarithm for a product, transforming it into using the difference of squares to further simplify.

Once the expression inside the logarithm is simplified using algebraic manipulation, you need to fully calculate the remaining straightforward arithmetic operations. After applying difference of squares, you calculate:

• \( (x^2)^2 - (\sqrt{x^4 + 1})^2 = x^4 - x^4 - 1 = -1 \)

The crucial step is taking the absolute value to make sure the logarithm function behaves properly: \( \ln| -1 | = \ln(1) \). Therefore, log equations reach their simplest form by considering such simplifications.

Real Numbers

When working with logarithms and algebraic expressions, understanding the set of real numbers is essential. Real numbers include all rational and irrational numbers, making the set very comprehensive and suitable for our calculations.

In the exercise, we note that the equation involving natural logarithms holds for any real number \(x\). Since real numbers are extensive, they cover zero, positive, and negative numbers without any gaps.
This is significant because it often influences the nature and solution of logarithmic expressions, which can only be defined for positive arguments. That’s why absolute values \(|x|\) are used, ensuring the properties of logarithms are satisfied for all real inputs. Thus, understanding the domain helps us correctly apply mathematical properties and identities.