Key Concepts and Formulas

• Graphing $y = \ln x$ and its transformations: Understanding the basic logarithmic function and how absolute values affect the graph.
• Area between curves: The area between two curves $f(x)$ and $g(x)$ from $a$ to $b$ is given by $\int_a^b |f(x) - g(x)| dx$.
• Properties of Logarithms: $\ln|x|$ is defined for all $x \ne 0$.

Step-by-Step Solution

Step 1: Analyze the functions and their graphs

We have four functions:

1. $y = \ln x$
2. $y = \ln |x|$
3. $y = |\ln x|$
4. $y = |\ln |x||$

Let's analyze each function and its graph.

• $y = \ln x$: This is the standard natural logarithm function, defined for $x > 0$.

• $y = \ln |x|$: Since $|x|$ is always non-negative, this function is defined for all $x \ne 0$. For $x > 0$, $y = \ln x$. For $x < 0$, $y = \ln(-x)$. The graph is symmetric about the y-axis.

• $y = |\ln x|$: This function is defined for $x > 0$. It takes the absolute value of $\ln x$. So, when $\ln x < 0$, i.e., $0 < x < 1$, the graph is reflected about the x-axis.

• $y = |\ln |x||$: This function is defined for all $x \ne 0$. It takes the absolute value of $\ln |x|$. For $x>0$, $y = |\ln x|$, and for $x<0$, $y = |\ln (-x)|$. The graph is symmetric about the y-axis.

Step 2: Find the intersection points and the region of interest

We want to find the area bounded by these four curves. Since $y = \ln x$ is only defined for $x > 0$, and the other functions are also defined for $x < 0$, and symmetric about the y-axis, we can find the area in the first quadrant ($x > 0$) and then multiply by 2 to get the total area.

For $x > 0$, we have:

1. $y = \ln x$
2. $y = \ln x$
3. $y = |\ln x|$
4. $y = |\ln x|$ This means the first two functions are identical for $x > 0$, and the last two are identical for $x > 0$.

The region is bounded by $y = \ln x$ and $y = |\ln x|$. We need to consider two cases: $0 < x < 1$ and $x > 1$.

Step 3: Calculate the area for $0 < x < 1$

In this interval, $\ln x < 0$, so $|\ln x| = -\ln x$. The area between the curves is given by: $A_1 = \int_0^1 (|\ln x| - \ln x) dx = \int_0^1 (-\ln x - \ln x) dx = \int_0^1 -2\ln x dx$ However, since $\ln x$ is not defined at $x=0$, we need to take a limit: $A_1 = \lim_{a \to 0^+} \int_a^1 -2\ln x dx$ We integrate by parts. Let $u = \ln x$, $dv = dx$, so $du = \frac{1}{x}dx$, $v = x$. $\int \ln x dx = x\ln x - \int x \cdot \frac{1}{x} dx = x\ln x - \int dx = x\ln x - x + C$ Therefore, $A_1 = \lim_{a \to 0^+} -2[x\ln x - x]_a^1 = \lim_{a \to 0^+} -2[(1\ln 1 - 1) - (a\ln a - a)] = \lim_{a \to 0^+} -2[(0 - 1) - (a\ln a - a)]$ We know that $\lim_{a \to 0^+} a\ln a = 0$ and $\lim_{a \to 0^+} a = 0$. Therefore, $A_1 = -2[-1 - (0 - 0)] = -2(-1) = 2$

Step 4: Calculate the area for $x > 1$

In this interval, $\ln x > 0$, so $|\ln x| = \ln x$. The area between the curves is given by: $A_2 = \int_1^e (\ln x - \ln x) dx = \int_1^e 0 dx = 0$ Since we are looking for the area bounded by these curves, we should consider the region where the graphs differ. The graphs of $y = \ln x$ and $y=|\ln x|$ differ for $0<x<1$. We already calculated the area in the first quadrant as $A_1 = 2$.

Step 5: Account for symmetry

Since the functions $y=\ln|x|$ and $y=|\ln|x||$ are symmetric with respect to the y-axis, and $y=\ln x$ and $y=|\ln x|$ are defined for $x>0$, the total area is twice the area we found in the interval $0 < x < 1$. Specifically, we need to calculate the area between $\ln|x|$ and $|\ln|x||$ from $-1$ to $0$. Due to the symmetry, this will be the same as the area from $0$ to $1$ between $\ln x$ and $|\ln x|$.

The total area is $2 \times A_1 = 2 \times 2 = 4$.

Common Mistakes & Tips

• Remember to consider the absolute value when calculating the area. Make sure you are integrating the difference between the curves, taking the absolute value of the difference if necessary to ensure the area is positive.
• Be careful with the limits of integration. The logarithmic function is only defined for positive arguments.
• When dealing with absolute values, split the integral into regions where the expression inside the absolute value is positive or negative.

Summary

We analyzed the four functions, identified the region bounded by them, and used integration to find the area. Due to the symmetry of the functions involving $|x|$, we calculated the area in the first quadrant and multiplied by 2 to get the total area. The area between the curves $y = \ln x$ and $y = |\ln x|$ for $0<x<1$ is 2. Accounting for the symmetry about the y-axis, the total area is 4.

Final Answer The final answer is \boxed{4}, which corresponds to option (A).