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The domain of a function refers to all the potential input values (x-values) that it can accept without causing any mathematical errors. For logarithmic functions, understanding the domain is crucial since not all numbers are valid inputs. The standard logarithmic function, written as \(f(x) = \ln x\), only accepts positive values for \(x\). That's because the logarithm of zero or any negative number is undefined. Hence, the domain of \(\ln x\) is \(x > 0\).

When we modify the function to \(f(x) = \ln 2x\), we still need \(2x > 0\). Dividing both sides of the inequality by 2, we determine \(x > 0\). Thus, the domain remains as all positive numbers \((0, \infty)\).

This domain ensures that for any input within this set, the output or y-value will be a real number. It is essential for the graph's consistency and functionality.

The natural logarithm, denoted as \(\ln\), is a specific logarithm with the base \(e\), where \(e\) is an irrational and transcendental number approximately equal to 2.71828. The function \(f(x) = \ln x\) maps positive real numbers into all real numbers.

Key characteristics of \(\ln x\) include:

• Intersects the x-axis at the point (1,0), since \(\ln 1 = 0\).
• Monotonically increasing, meaning it steadily rises and never decreases for \(x > 0\).
• Has a vertical asymptote at \(x = 0\) because as x approaches zero from the positive side, \(\ln x\) approaches negative infinity.

For the function \(f(x) = \ln 2x\), it similarly behaves like \(\ln x\), but with modifications due to the multiplier (factor of 2).This compression factor means that as \(x\) increases, the rate at which \(\ln 2x\) increases is faster than that of \(\ln x\). But fundamentally, it retains the growth characteristics of natural logarithms.

Transformation of a function involves modifying it in a specific way to achieve a desired change in its graph's appearance or behavior. The common transformations include:

• Translations (shifting the graph left, right, up, or down)
• Reflections (flipping the graph over a line, such as the x-axis or y-axis)
• Stretching or compressing (changing the graph's width or height)

In the function \(f(x) = \ln 2x\), a transformation occurs where the base function \(\ln x\) is horizontally compressed by a factor of 2. This means the graph will be steeper than that of \(\ln x\).

When sketching or analyzing such transformations, it is important to note the impact on key points and asymptotes. Here, \(\ln 2x\) will cross the x-axis at \((1,0)\) like \(\ln x\), but will rise more sharply.

This specific compression makes it crucial to adjust plotting points and understand that every value on the graph now corresponds to input values twice as small, compared to the basic \(\ln x\). This knowledge aids in accurately sketching and interpreting the graph's new behavior.