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The natural logarithm, often denoted as \( \ln \), is a logarithm with the base \( e \), where \( e \) is an irrational and transcendental number approximately equal to 2.71828. The natural logarithm of a positive number \( t \) symbolizes the power to which \( e \) must be raised to equal \( t \). Mathematically, this is expressed as: - \( e^{\ln(t)} = t \). Understanding the natural logarithm is pivotal in calculus, especially when dealing with integrals of functions involving \( \frac{1}{t} \). The integral of \( \frac{1}{t} \) is \( \ln|t| \). This fact is essential when computing definite integrals that lead to expressions such as \( \ln y - \ln x \).
When evaluating expressions like \( \ln|y| - \ln|x| \), remember that it can also be rewritten as a single logarithm: - \( \ln\left(\frac{|y|}{|x|}\right) \). This property aids in simplifying the computation of definite integrals that result from functions like \( \frac{1}{t} \).

Function evaluation refers to the process of determining the value of a function at particular inputs, typically numbers. In the context of calculus, this often involves evaluating integrals. For example, to evaluate \( g(x, y) = \int_{x}^{y} \frac{1}{t} dt \), we first must understand what transformations take place from the integral to a computable result.
The integral \( \int \frac{1}{t} dt \) yields \( \ln|t| \). Therefore, the evaluation of the function \( g(x, y) \) becomes \( \ln|y| - \ln|x| \). This evaluation directly involves applying the limits \( x \) and \( y \) into the expression.

• For \( g(4,1) \), substitute \( x = 4 \) and \( y = 1 \), leading to \( \ln|1| - \ln|4| = -\ln4 \).
• For \( g(6,3) \), substitute \( x = 6 \) and \( y = 3 \), leading to \( \ln|3| - \ln|6| = -\ln2 \).

This step not only involves inserting values but also simplification based on the properties of logarithms.

Solving calculus problems often involves a strategic approach. Let's break down the process using the integral \( \int_{x}^{y} \frac{1}{t} dt \) as an example. The primary goal is to evaluate the integral over a defined interval, in this case, from \( x \) to \( y \).
Here are the general steps involved:

• Recognize the function to integrate: Identify that the function \( \frac{1}{t} \) integrates to \( \ln|t| \).
• Understand and apply limits of integration: Substitute the upper and lower limits into the result of the integral. This results in evaluation at \( t = y \) and \( t = x \).
• Simplify: Utilize properties such as \( \ln|y| - \ln|x| = \ln\frac{|y|}{|x|} \) for simplification.

Through such steps, you arrive at a concrete solution, exemplifying how calculus is not just about calculations, but also involves logical reasoning and simplification of complex expressions.
By following the structured approach, even more complex integrals and functions can be tackled effectively.