Understanding the Range of y ln(x) - 2 and the Natural Logarithm

The natural logarithm, denoted as ln(x), is a fundamental mathematical function. This article aims to explore the range of the function y ln(x) - 2 in detail. We will delve into the properties of the natural logarithm and the transformations involved.

Properties of the Natural Logarithm Function

The natural logarithm function, ln(x), is defined for positive real numbers (i.e., x > 0). It is a logarithmic function with base e (Euler's number), approximately equal to 2.71828. The function is both continuous and differentiable over its domain, and it has the following properties:

Its domain is (0, ∞). Its range is all real numbers, denoted as (-∞, ∞). The graph of ln(x) increases without bound as x increases. At x 1, the value of ln(x) is 0. ln(x) is an increasing function.

Understanding the Function y ln(x) - 2

Next, let's consider the function y ln(x) - 2. This function involves a vertical shift of the graph of ln(x) downward by 2 units. Essentially, it takes the natural logarithm of x and then subtracts 2 from the result.

Domain of y ln(x) - 2

The domain of the function y ln(x) - 2 is the same as the domain of ln(x). Therefore, the domain of y ln(x) - 2 is (0, ∞). This means that for any value of x 0, the function ln(x) - 2 is defined.

Range of y ln(x) - 2

To understand the range of y ln(x) - 2, let's analyze how the shift affects the range of ln(x).

Original Function y ln(x)

The range of the natural logarithm function, ln(x), is all real numbers, meaning ln(x) can take any value in (-∞, ∞). To visual this, imagine the graph of ln(x), which covers the entire vertical axis from negative infinity to positive infinity.

Vertical Shift Downward

When we apply the transformation y ln(x) - 2, we shift the graph of ln(x) downward by 2 units. This means that every output value of ln(x) is decreased by 2. Consequently, the range of ln(x) - 2 is shifted downward by 2 units as well.

Since the original range of ln(x) is (-∞, ∞), shifting it downward by 2 units results in the new range being (-∞ - 2, ∞ - 2) (-∞, ∞). Therefore, the range of y ln(x) - 2 is also all real numbers, (-∞, ∞).

Key Points and Conclusion

In summary, the function y ln(x) - 2 retains the domain of (0, ∞) and the range of all real numbers, (-∞, ∞). Key points to remember:

The domain of y ln(x) - 2 is (0, ∞). The range of y ln(x) - 2 is (-∞, ∞). The graph of y ln(x) - 2 is the same as the graph of ln(x), but shifted downward by 2 units.

Understanding these properties and transformations is crucial in analyzing and working with the natural logarithm function and its variations.