Problem 10 - Thomas Calculus

Question

Rewrite the expressions in Exercises \(5-10\) in terms of exponentials and simplify the results as much as you can. $$\ln (\cosh x+\sinh x)+\ln (\cosh x-\sinh x)$$

Step-by-Step Solution

Answer

The simplified expression is 0.

Step 1: Recognize the use of Logarithm Properties

The given expression is a sum of two logarithms: \( \ln(\cosh x + \sinh x) + \ln(\cosh x - \sinh x) \). We can use the property of logarithms which states that the sum of logs can be rewritten as the log of a product: \( \ln(a) + \ln(b) = \ln(a \cdot b) \).

Step 2: Apply the Logarithm Property

Applying the property, we have \( \ln((\cosh x + \sinh x) \cdot (\cosh x - \sinh x)) \). This simplifies to \( \ln((\cosh^2 x - \sinh^2 x)) \) using the difference of squares.

Step 3: Use Hyperbolic Identity

Apply the hyperbolic trigonometric identity \( \cosh^2 x - \sinh^2 x = 1 \). Therefore, the expression simplifies to \( \ln(1) \).

Step 4: Simplify the Expression

The logarithm of 1 is 0, so \( \ln(1) = 0 \). This is the simplest form of the expression.

Key Concepts

Understanding Hyperbolic Functions

Hyperbolic functions, much like their trigonometric counterparts, are important in various fields, including calculus and complex analysis. They are related to the exponential function and have some similarities to the sine and cosine functions. The two primary hyperbolic functions are the hyperbolic sine and hyperbolic cosine, denoted as \( \sinh x \) and \( \cosh x \) respectively.

These functions are defined as follows:

• \( \sinh x = \frac{e^x - e^{-x}}{2} \)
• \( \cosh x = \frac{e^x + e^{-x}}{2} \)

These definitions show that hyperbolic functions are based on the exponential function \( e^x \). They possess unique properties and identities that are quite useful, especially when simplifying expressions.

For example, a fundamental hyperbolic identity is \( \cosh^2 x - \sinh^2 x = 1 \). This identity is analogous to the Pythagorean identity in trigonometry \( \cos^2 x + \sin^2 x = 1 \), which highlights the relationship between \( \cosh x \) and \( \sinh x \). Knowing these identities helps in simplifying and transforming expressions that include hyperbolic functions.

Mastering Simplifying Expressions

In mathematics, simplifying expressions involves reducing and transforming them into their simplest form. This process often makes them easier to work with, especially in complex calculations or algebraic manipulations.

Using identities like the difference of squares or hyperbolic identities greatly aids in simplification. For instance, in our original exercise, by recognizing the expression \( (\cosh x + \sinh x)(\cosh x - \sinh x) \), we used the identity for a difference of squares:

• \((a+b)(a-b) = a^2 - b^2\)

Applying it here gives us \( \cosh^2 x - \sinh^2 x \).

Additionally, applying known identities like \( \cosh^2 x - \sinh^2 x = 1 \) is crucial because it reduces complexity and can transform the expression into a much simpler numeric result. Simplifying not only helps in achieving a simpler form but also enhances understanding of the relationships between various mathematical components.

Exploring Properties of Logarithms

Logarithms play a crucial role in simplifying complex expressions and solving equations. One of their fundamental properties is that they convert multiplication into addition and vice versa. In our example, we used this property where the sum of two logs is equivalent to the logarithm of their product.

• The property can be stated as: \( \ln(a) + \ln(b) = \ln(a \cdot b) \)
• Another critical property is: \( \ln\left(\frac{a}{b}\right) = \ln a - \ln b \)

These properties help in breaking down logs into more manageable parts or combining them to result in a simpler expression.

In the original exercise, this property allowed us to transform the initial expression into a single logarithmic expression: \( \ln((\cosh x + \sinh x) \cdot (\cosh x - \sinh x)) \).

The expression then became simpler by applying further identities. Moreover, knowing that the logarithm of 1 is 0 is another vital piece of information which underscores that some logarithmic expressions can resolve to much simpler forms. This understanding is vital across different mathematical problems involving logarithms.