91Ó°ÊÓ

The natural logarithm is a special type of logarithm that uses the base \(e\), where \(e\) is an irrational and transcendental number approximately equal to 2.71828. It is commonly denoted as \(\ln\). Understanding \(\ln\) is crucial because it frequently appears in calculus, complex analysis, and many real-world applications such as growth processes.

The natural logarithm has several important properties:

• \(\ln(1) = 0\) because \(e^0 = 1\).
• \(\ln(e) = 1\) because \(e^1 = e\).
• The function \(\ln(x)\) is only defined for \(x > 0\).
• It is the inverse of the exponential function \(e^x\).

These properties make the natural logarithm particularly useful in solving problems involving exponential growth or decay.

In our original exercise, we encounter the term \(\ln(\frac{1}{2})\). Calculating this involves understanding that it means "what power must \(e\) be raised to, to get \(\frac{1}{2}\)?"

Function evaluation entails substituting a given input into a function to calculate the corresponding output. This process allows one to determine the value of the function at specific points.

Functions generally have the form \(f(x)\), where \(x\) is the input. To evaluate the function, simply replace \(x\) with the specified input value. Here are some simple steps to evaluate a function:

• Identify the function and the input variable. In our example, \(g(x) = -\ln x\) with the input \(x = \frac{1}{2}\).
• Substitute the input variable with the specified value: \(g\left(\frac{1}{2}\right) = -\ln\left(\frac{1}{2}\right)\).
• Compute the result based on the given distinction or operation within the function. Here, compute \(-\ln\left(\frac{1}{2}\right)\).

Evaluating the function correctly is vital to understanding what the function represents at that particular value.

Substitution is a straightforward technique often used in mathematics to replace a variable in an expression with a numerical value or another expression. When working with functions, substitution can help us simplify the expressions and solve for specific values.

In our example, substitution involves replacing \(x\) in the function \(g(x) = -\ln x\) with \(\frac{1}{2}\).
Here's a simple approach:

• Write the function initially: \(g(x) = -\ln x\).
• Perform the substitution by replacing \(x\) with the given value, so it transforms to \(g\left(\frac{1}{2}\right) = -\ln\left(\frac{1}{2}\right)\).
• Once substituted, proceed with any required mathematical operations to solve the expression.

This process is essential in various mathematical tasks as it simplifies complex expressions by substituting known values or equations.