Let's break this down step by step.

(a) Finding the domain of $f(x) = 2 \ln(2x - 1) + 4$

The logarithmic function $\ln(x)$ (or $\log_e(x)$) is only defined when its argument is positive, i.e., $x > 0$.

For the function $f(x) = 2 \ln(2x - 1) + 4$, we need the expression inside the logarithm, $2x - 1$, to be greater than 0:

$2x - 1 > 0$

Solving this inequality:

$2x > 1 \quad \Rightarrow \quad x > \frac{1}{2}$

Thus, the domain of $f(x)$ is:

$\boxed{x > \frac{1}{2}}$

(b) Describing the transformations

The basic logarithmic function is $y = \ln(x)$, and we need to describe the transformations applied to this function to obtain $f(x) = 2 \ln(2x - 1) + 4$.

Let’s identify each transformation step by step:

1. Horizontal scaling:

   • The term $\ln(2x - 1)$ involves $2x - 1$. First, consider the factor of 2 in front of $x$. This causes a horizontal compression by a factor of $\frac{1}{2}$ because for $\ln(kx)$, the graph is compressed horizontally by $k$.

2. Horizontal translation:

   • The $-1$ in $2x - 1$ indicates a horizontal shift to the right by $\frac{1}{2}$. This is because solving $2x - 1 = 0$ gives $x = \frac{1}{2}$, meaning the graph is shifted right by $\frac{1}{2}$.

3. Vertical scaling:

   • The factor of 2 in front of $\ln(2x - 1)$ causes a vertical stretching of the graph by a factor of 2.

4. Vertical translation:

   • The $+4$ at the end indicates a vertical shift upwards by 4 units.

Final set of transformations:

• Horizontal compression by a factor of $\frac{1}{2}$.
• Horizontal shift right by $\frac{1}{2}$.
• Vertical stretch by a factor of 2.
• Vertical shift up by 4 units.

These transformations, when applied to the graph of $y = \ln(x)$, result in the graph of $f(x) = 2 \ln(2x - 1) + 4$.

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Would you like further details or have any questions about these transformations? Here are 5 related questions you might find useful:

1. How would you find the range of $f(x)$?
2. What would be the effect on the domain if the constant inside the logarithm was different, say $\ln(3x - 1)$?
3. How would the transformations change if the function was $3 \ln(2x - 1) + 2$?
4. Can you graph $f(x) = 2 \ln(2x - 1) + 4$ and describe its key features (intercepts, asymptotes)?
5. How would the graph behave as $x$ approaches infinity?

Tip: Always ensure the argument inside the logarithmic function is positive when finding domains involving logarithms.