Problem 37 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 1-38: Find all solutions of the equation. $$ \cos (\ln x)=0 $$

Step-by-Step Solution

Answer

The solutions are \( x = e^{\frac{(2n+1)\pi}{2}} \) for integer \( n \).

Step 1: Understanding the Equation

The given equation is \( \cos(\ln x) = 0 \). We want to find the values of \( x \) for which the cosine of the natural logarithm of \( x \) is zero. Recall that the cosine function is zero at every odd multiple of \( \frac{\pi}{2} \), i.e., \( \frac{(2n+1)\pi}{2} \), where \( n \) is an integer.

Step 2: Equating Argument of Cosine to Zeros

Set the argument of the cosine function, which is \( \ln x \), equal to the values that make cosine zero: \( \ln x = \frac{(2n+1)\pi}{2} \) for integer \( n \).

Step 3: Solve for x

To solve for \( x \), exponentiate both sides of the equation from Step 2 since \( \ln x = \frac{(2n+1)\pi}{2} \). Therefore, \( x = e^{\frac{(2n+1)\pi}{2}} \). This equation gives the solutions for \( x \).

Step 4: State Solutions in General Form

The solutions to the equation \( \cos (\ln x) = 0 \) are \( x = e^{\frac{(2n+1)\pi}{2}} \) for every integer \( n \). This represents a sequence of values of \( x \) that satisfies the original equation.

Key Concepts

Cosine Function

The cosine function is one of the fundamental trigonometric functions which is pivotal in trigonometry. It often appears as part of periodic phenomena like waves. The cosine of an angle in a right-angled triangle is the ratio of the length of the adjacent side to the hypotenuse. This definition extends to the unit circle, where cosine represents the x-coordinate of a point on the circle.

Important properties of the cosine function include:

• Its range is between \[ -1 \] and \[ 1 \].
• Its period is \[ 2\pi \], meaning that the function repeats its values every \[ 2\pi \] radians.
• It is an even function, so \[ \cos(-x) = \cos(x) \].

One key characteristic is that cosine is zero at odd multiples of \[ \frac{\pi}{2} \]. This means the function crosses the x-axis at \[ \frac{(2n+1)\pi}{2} \], where \( n \) is any integer. Understanding this behavior helps solve equations involving the cosine function.

Natural Logarithm

The natural logarithm, denoted as \( \ln(x) \), is a logarithm with base \( e \), where \( e \approx 2.718 \). The natural logarithm has a variety of applications in both mathematics and science, particularly in scenarios where exponential growth or decay is involved.

Key features of the natural logarithm include:

• \( \ln(1) = 0 \)
• It is undefined for \( x \leq 0 \).
• It is monotonically increasing, meaning it always increases as x increases.
• The inverse function of the natural logarithm is the exponential function \( e^x \).

Using natural logarithms simplifies the manipulation of exponential equations, especially when combined with trigonometric functions. For example, to solve \( \cos(\ln x) = 0 \), understanding \( \ln x \) allows us to solve for \( x \) through exponential operations.

Integer Solutions

Integer solutions refer to solutions of an equation that are whole numbers, including positive, negative numbers, and zero. In the context of equations involving periodic functions like cosine, it's common to see solutions characterized by integer multiples of certain values.

In trigonometric contexts, solving \( \cos(\ln x) = 0 \) results in identifying the values where the argument of cosine is an odd multiple of \( \frac{\pi}{2} \). Here, the integer \( n \) represents these multiples:

• \( n = 0 \) yields the first zero point.
• \( n = 1, -1, 2, -2, \ldots \) yield subsequent points.

Thus, integer \( n \) helps in expressing an infinite sequence of solutions, which, when substituted into exponential equations, generate real-number solutions for \( x \).

Exponential Functions

Exponential functions involve expressions of the form \( a^x \), where \( a \) is a constant base greater than zero. Of special interest is the base being Euler's number \( e \), forming the natural exponential function \( e^x \), a fundamental concept in calculus and continuous growth models.

Some features:

• Exponential functions grow rapidly, increasing or decreasing by large factors over intervals.
• The function \( e^x \) has the intriguing property that its derivative is itself.
• It serves as the inverse of the natural logarithm, making it crucial for solving equations involving logarithms.

In solving \( \ln x = \frac{(2n+1)\pi}{2} \), we employ the exponential function to solve for \( x \). By exponentiating both sides, we calculate \( x \) as \( e^{\frac{(2n+1)\pi}{2}} \), providing clarity on how exponential functions enable us to solve logarithmic equations.