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The natural logarithm, denoted as \(\text{ln}\), is the logarithm to the base of the mathematical constant \(\text{e}\). The constant \(\text{e}\) is approximately equal to 2.718281828. The natural logarithm of a number \(\text{x}\), written as \(\text{ln}(x)\), represents the power to which \(\text{e}\) must be raised to obtain the number \(\text{x}\). For example, if \(\text{e}^y = x\), then \(\text{ln}(x) = y\). Here are some key points to understand:

• The natural logarithm is used in many areas of mathematics and science, especially in solving exponential equations.
• It is the inverse function of the exponential function.
• Properties include \( \text{ln}(1) = 0 \), \( \text{ln}(e) = 1 \), and \( \text{ln}(ab) = \text{ln}(a) + \text{ln}(b) \).

In the context of the exercise, after solving the quadratic equation, we take the natural logarithm of \(3 + 2\sqrt{2}\)\ to express the final answer in terms of \(\text{ln}\). This shows the connection between exponential equations and logarithms.

The hyperbolic cosine function, represented as \(\text{cosh}(x)\), is one of the hyperbolic functions and is analogous to the trigonometric cosine function. It is defined by the formula: \[ \text{cosh}(x) = \frac{e^x + e^{-x}}{2} \]
Here are some key characteristics of hyperbolic cosine:

• Like the trigonometric functions, hyperbolic functions are used in various areas of mathematics, including calculus and complex analysis.
• The hyperbolic functions have exponential representation, making them useful in solving differential equations and integrals.
• The hyperbolic cosine function is even, meaning \( \text{cosh}(-x) = \text{cosh}(x) \).

In this exercise, we use the definition of \(\text{cosh}(y)\) to set up an equation in terms of exponentials and then solve for \(y\) by working through the equation step by step.

A quadratic equation is a second-order polynomial equation in a single variable \(x\), with the general form: \[ ax^2 + bx + c = 0 \]
Here are some essential concepts about quadratic equations:

• The solutions to the quadratic equation are given by the quadratic formula: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
• The term \(b^2 - 4ac\) is known as the discriminant. It determines the nature of the roots (real and distinct, real and repeated, or complex).
• Quadratic equations can be solved by factoring, completing the square, or using the quadratic formula.

In this exercise, we treat \(e^y\) as a variable and rearrange the equation to form a quadratic equation: \( e^{2y} - 6e^y + 1 = 0 \). Solving this quadratic equation allows us to find \(e^y\), which then helps to express \(y\) in terms of the natural logarithm.