91Ó°ÊÓ

Logarithmic functions are a crucial part of mathematics, especially in calculus and algebra. A basic logarithm function is written as \(y = \log_b(x)\), where \(b\) is the base and \(x\) is the argument. The logarithm tells us what power we need to raise \(b\) to get \(x\). For example, with base 10: \(\log_{10}(100) = 2\) means 10 raised to the power of 2 is 100.

If we consider the power rule for logarithms, this rule provides a way to simplify the logarithm of a power of a number. The power rule states: \(\log_b(x^a) = a \cdot \log_b(x)\). This transformation allows us to move the exponent in front of the logarithm for easier computation and simplification.

In the given exercise, we start with the function \(y = \log(x^2)\). By applying the power rule, this transforms into \(y = 2 \cdot \log(x)\). While this makes the expression simpler, it's critical to check if these functions are identical in all aspects, particularly their domains.

The domain of a function is the set of all possible input values (x-values) that the function can accept. For logarithmic functions, the argument of the logarithm must be greater than zero because the logarithm of a non-positive number is undefined.

Let's look at the domains of our two functions from the exercise:

• For \(y = \log(x^2)\), the argument \(x^2\) must be positive. Since \(x^2 > 0\) for all \(x eq 0\), the domain is \(x \in (-\infty, 0) \cup (0, \infty)\).
• For \(y = 2 \cdot\ \log(x)\), the argument \(x\) must be positive. Therefore, the domain is \(x \in (0, \infty)\).

This shows that while the power rule transformation appears straightforward, it can change the domain significantly. The original function is defined for all non-zero x-values, but after applying the power rule, the new function is only defined for positive x-values.

Consider another example: the function \(y = \log(x(x-1))\) has a domain where \(x(x-1) > 0\). Solving this inequality, we find the domain is \(x \in (-\infty, 0) \cup (1, \infty)\). If we tried to simplify this function using logarithm rules, it would reveal how the domain changes under different transformations.

Graphing functions is a powerful way to visualize their behavior and understand their properties. Let's examine our functions:

• Graph of \(y = \log(x^2)\): This function is symmetric about the y-axis because \(x^2\) is always non-negative and it works similarly for positive and negative values of \(x\). The graph is undefined at \(x = 0\).
• Graph of \(y = 2 \cdot\ \log(x)\): This function is only defined for positive x-values. The graph starts from \(x = 0+\) and continues to stretch upward and to the right. It does not exist for negative x-values.

By comparing both graphs, we see how the domain affects the appearance and behavior of the functions. The original function's graph is wider and exists on both positive and negative sides of the x-axis, while the transformed function is narrower and only on the positive side.

Understanding the graphical representation of functions alongside their algebraic expressions deepens comprehension. It visually demonstrates why certain operations like the power rule change the domain and behavior of functions in significant ways.