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The domain of a function refers to all possible input values (typically represented as "x") that allow the function to be defined. For the function, it is crucial to understand which x-values we can plug into the equation without making it undefined or "break."

For logarithmic functions, such as the natural logarithm, the domain is limited to positive values because compute values without errors. For instance, negative values or zero would make the logarithm undefined, as calculating the logarithm of zero or a negative number does not yield real numbers.

• For the function \(f(x) = \ln x\), the domain is \(x > 0\).
• Similarly, for the function \(f(x) = 3 \ln x\), the domain remains \(x > 0\) since multiplying by a constant does not change the valid input range.

Understanding the domain is key when graphing or solving equations, ensuring that we only consider values that keep the function within its valid boundaries.

A vertical stretch occurs when every y-coordinate of a function is multiplied by a constant factor. This transformation affects how "tall" or "short" the function graph appears without altering its width.

For the function \(f(x) = 3 \ln x\), the natural logarithm function \(\ln x\) is stretched vertically by a factor of 3. This means that each point on the original \(\ln x\) graph is pushed further away from the x-axis, making the graph steeper and more pronounced.

• If a point \((a, b)\) is on the graph of \(\ln x\), it transforms into \((a, 3b)\) on the graph of \(3 \ln x\).
• This vertical stretch does not affect the domain. The x-values remain the same, only affecting the y-values.

This concept is important when sketching the graph, as it indicates how the graph behaves compared to the original function. It visually demonstrates the influence of constants on a logarithmic function.

The natural logarithm, denoted as \(\ln(x)\), is a specific logarithm to the base \(e\), where \(e\) is approximately equal to 2.71828. It is widely used in mathematics due to its natural properties and simplifications when dealing with exponential and logarithmic calculations.

Key characteristics of the natural logarithm include:

• Continuous and increasing behavior for all values where \(x > 0\).
• It crosses the y-axis at the point (1,0) since \(\ln(1) = 0\).
• Approaches infinity as \(x\) increases and approaches zero as \(x\) approaches zero from the positive side.
• The natural logarithm is undefined for \(x \le 0\).

The natural logarithm is foundational in understanding logarithmic and exponential growth and decay, and it serves as the starting point for transformations like vertical stretches, as seen in functions such as \(3 \ln x\). By grasping the behavior and unique aspects of \(\ln(x)\), we can better understand its applications in calculus and beyond.