Problem 37 - Precalculus: Functions and Graphs

Question

Find all solutions of the equation. $$\cos (\ln x)=0$$

Step-by-Step Solution

Answer

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

Step 1: Understand the Equation

The given equation is \( \cos(\ln x) = 0 \). Our job is to find the values of \( x \) that satisfy this equation. We know that \( \cos(\theta) = 0 \) when \( \theta = \frac{\pi}{2} + k\pi \), where \( k \) is an integer. Thus, we need to find \( x \) such that \( \ln x = \frac{\pi}{2} + k\pi \).

Step 2: Solve for \( x \)

Starting from \( \ln x = \frac{\pi}{2} + k\pi \), we solve for \( x \) by exponentiating both sides: \( e^{\ln x} = e^{\frac{\pi}{2} + k\pi} \), which simplifies to \( x = e^{\frac{\pi}{2} + k\pi} \).

Step 3: Conclude the Solutions

The solutions to the equation \( \cos (\ln x) = 0 \) are given by \( x = e^{\frac{\pi}{2} + k\pi} \) for integer values of \( k \). These represent the infinite set of solutions where each value corresponds to a different \( k \).

Key Concepts

Trigonometric Functions

Trigonometric functions are mathematical functions that relate angles of triangles to the lengths of sides in a right triangle. The most common trigonometric functions are sine (\(\sin\theta\)), cosine (\(\cos\theta\)), and tangent (\(\tan\theta\)). These functions are fundamental in studying periodic phenomena, like waves and oscillations.

In the equation \(\cos(\ln x) = 0\), we specifically deal with the cosine function, which equals zero at certain points. The cosine function is periodic with a period of \(2\pi\).

• \(\cos\theta = 0\) at angles \(\theta = \frac{\pi}{2} + k\pi\), where \(k\) is any integer.

This periodic property helps us find when the natural logarithm of \(x\) equals one of those angles to satisfy the equation.

Natural Logarithm

The natural logarithm, denoted as \(\ln\), is the logarithm to the base \(e\), where \(e\) is an irrational number approximately equal to 2.71828. It is widely used in mathematical calculations, especially in calculus and when dealing with continuous growth or decay processes.

The natural logarithm transforms multiplicative processes into additive ones, which is crucial in solving equations like \(\cos(\ln x) = 0\). In this equation, we need to find when the \(\ln x\) value results in cosine equalling zero.

• Logarithms change multiplication into addition: \(\ln(a \times b) = \ln a + \ln b\).
• Exponentiating cancels out the logarithm, as seen with: \(e^{\ln x} = x\).

Utilizing these properties, we convert from the logarithmic form to potential solutions when aligned with cosine's zero points.

Exponential Functions

Exponential functions involve expressions where the variable appears in the exponent. A basic form is \(y = a^{x}\), but the most important one in advanced mathematics is \(e^{x}\). The function \(e^{x}\) is unique because its derivative is the same as the function itself; \(\frac{d}{dx}(e^{x}) = e^{x}\).

In solving our given equation \(\cos(\ln x) = 0\), we found that exponentiating helped convert the natural logarithm back to its number form.

• Starting from \(\ln x = \frac{\pi}{2} + k\pi\), we express it in terms of \(x\) by exponentiating: \(e^{\ln x} = e^{\frac{\pi}{2} + k\pi}\).
• This simplifies to \(x = e^{\frac{\pi}{2} + k\pi}\), offering a set of potential solutions as \(k\) varies among all integers.

Exponential functions provide a way to transform logarithmic equations back into their original numeric settings, offering clarity in complex solutions like ours.