Problem 10 - Thomas Calculus in SI Units

Question

Rewrite the expressions 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 expression simplifies to 0.

Step 1: Apply Logarithm Product Rule

Start by using the logarithm property \( \ln(a) + \ln(b) = \ln(a \cdot b) \). In this case, apply it to the given expression: \[ \ln(\cosh x + \sinh x) + \ln(\cosh x - \sinh x) = \ln((\cosh x + \sinh x)(\cosh x - \sinh x)) \]

Step 2: Simplify Using Hyperbolic Identity

Recognize the identity for hyperbolic functions: \( \cosh^2 x - \sinh^2 x = 1 \). Use this identity to simplify the expression:\[ (\cosh x + \sinh x)(\cosh x - \sinh x) = \cosh^2 x - \sinh^2 x = 1 \]

Step 3: Simplify Logarithm of 1

Since \( \ln(1) = 0 \), the expression simplifies to:\[ \ln((\cosh x + \sinh x)(\cosh x - \sinh x)) = \ln(1) = 0 \]

Step 4: Final Simplification

There are no other terms or factors to simplify, so the final result of the given expression is 0.

Key Concepts

Exponential Functions

Exponential functions are fundamental in mathematics, especially when dealing with growth and decay models. They are functions of the form \[ f(x) = a \cdot b^x \] where:

• \(a\) is a constant, providing the initial quantity.
• \(b\) is the base of the exponential function.

Exponential functions show how a quantity increases rapidly due to repeated multiplication by a constant base.

In the context of logarithmic and hyperbolic functions, we often express hyperbolic identities through exponential terms. For example, the hyperbolic sine and cosine are defined by:

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

These formulations reveal the relationship between exponential functions and hyperbolic functions, essential in simplifying complex expressions like the one in our exercise.

Hyperbolic Identities

Hyperbolic identities are relations similar to trigonometric identities, but they involve hyperbolic functions like hyperbolic sine (\sinh) and hyperbolic cosine (\cosh). One core identity is \(\cosh^2 x - \sinh^2 x = 1\), analogous to the Pythagorean identity for trig functions.

The identity \((\cosh x + \sinh x)(\cosh x - \sinh x) = 1\) arises from this relationship. This identity is crucial in simplifying expressions, as seen in the original exercise. Using these identities helps reduce complex log functions into simpler forms, ultimately leading to easier calculations and interpretations.

Understanding these basic identities allows you to transform and alter expressions efficiently, exemplifying the synergy between algebraic simplification and hyperbolic functions.

Algebraic Simplification

Algebraic simplification involves the process of making an algebraic expression as simple as possible. This can involve factoring, expanding, or reducing expressions to reach a cleaner form.

In simplifying logarithmic expressions, such as the exercise provided, you can use logarithmic properties like the product, quotient, and power rules to combine and reduce expressions. For instance:

• The product rule states \(\ln(a) + \ln(b) = \ln(a \cdot b)\), useful for merging two logarithms into a single one.

Here's how these concepts apply to the given problem: By applying the product rule, two logarithmic terms become one. Then, knowing the hyperbolic identity \(\cosh^2 x - \sinh^2 x = 1\), you can simplify the result into \(\ln(1)\), which equals zero.

This method showcases how algebraic simplification can streamline complex logarithmic expressions, highlighting that smart application of rules leads to efficient problem-solving.