Problem 5 - Calculus Early Transcendentals: Pearson New International Edition

Question

Simplify the given expression. $$ \ln e^{\cos x} $$

Step-by-Step Solution

Answer

The simplified expression is \( \cos x \).

Step 1: Identify the Relationship

Recall that the natural logarithm and the exponential function are inverses, so \( \ln(e^z) = z \).Here, \( z = \cos x \), which means that simplifying \( \ln(e^{\cos x}) \) involves applying this property.

Step 2: Apply the Property

Apply the inverse relationship of natural logarithms and exponentials to the expression: \[ \ln(e^{\cos x}) = \cos x \] This means that exponentiation and the logarithm operation cancel each other out, leaving just \( \cos x \).

Step 3: Simplified Expression

The simplified expression of \( \ln(e^{\cos x}) \) is simply \( \cos x \). There is no further simplification needed for this expression.

Key Concepts

Exponential Functions

Exponential functions are a fundamental building block in mathematics, characterized by the function form \( f(x) = a^{x} \), where \( a \) is a constant termed the base, and \( x \) is the exponent. A common base is the mathematical constant \( e \), approximately equal to 2.71828, known as Euler's number.
\( e^x \) functions are prominent in many mathematical models, particularly growth and decay processes such as population growth or radioactive decay.
These functions have distinct properties:

• Exponential growth: When the base is greater than 1, like \( e^x \), the function demonstrates rapid growth.
• Exponential decay: When the base is between 0 and 1, the function decreases towards zero.
• Naturally linked to logarithms: Exponentials often pair with logarithmic functions, which help solve equations involving exponentials.

Understanding these functions is essential for interpreting their behavior and solving equations like the one in the original exercise.

Inverse Functions

Inverse functions are a key concept in mathematics, providing a way to "reverse" the effect of a function. If you apply a function and then its inverse, you'll get back to the original value. Mathematically, if \( f(x) \) is a function, its inverse \( f^{-1}(x) \) satisfies the condition:\[ f(f^{-1}(x)) = f^{-1}(f(x)) = x \]Inverse functions are important in many fields, as they allow us to find specific values for which a function produces a known output.
In the context of the exercise, we see this with the natural logarithm \( \ln(x) \) and the exponential function \( e^x \). These functions are inverses of each other, meaning:

• The natural logarithm of an exponential \( \ln(e^x) \) results in \( x \).
• Exponentiating a logarithm \( e^{\ln(x)} \) also results in \( x \).

This inverse relationship is key to simplifying expressions involving logarithms and exponentials, as demonstrated in the original problem where \( \ln(e^{\cos x}) \) simplifies directly to \( \cos x \).

Trigonometric Functions

Trigonometric functions model the relationships between angles and sides of right-angled triangles. They are especially useful in describing oscillations, waves, and other periodic phenomena. The primary trigonometric functions are sine (\( \sin \)), cosine (\( \cos \)), and tangent (\( \tan \)). Among these:

• Cosine (\( \cos(x) \)) gives the length of the adjacent side over the hypotenuse in a right triangle and appears frequently in various branches of mathematics.
• Sine (\( \sin(x) \)) provides the ratio of the opposite side to the hypotenuse.
• Tangent (\( \tan(x) \)) combines these ratios as \( \tan(x) = \frac{\sin(x)}{\cos(x)} \).

Cosine is crucial in problems involving transformations and periodic functions, making math solutions versatile and applicable to real-world scenarios.
In the exercise, cosine appears within an exponential function, showcasing the integration of trigonometric and exponential functions. Since the \( \ln \) and \( e \) operations cancel, recognizing \( \cos x \) as the simplified result illustrates the core role trigonometric functions play within complex expressions.