Problem 11 - Calculus of a Single Variable

Question

In Exercises 9–16, sketch the graph of the function and state its domain. $$ f(x)=\ln 2 x $$

Step-by-Step Solution

Answer

The domain of the function \(f(x) = \ln 2x\) is \(x>0\), and the graph of the function is a steeper version of the graph of \(f(x) = \ln x\), starting at the x-axis at x=1 and getting progressively steeper as x increases. The function is undefined for \(x\leq0\).

Step 1: Determine the Domain

The first step is to find the domain of \(f(x)=\ln 2x\). The argument of a logarithm, in this case, 2x, must be greater than zero so we derive an inequality \(2x>0\). In solving, we find that \(x>0\). So, the domain of the function is \(x\in(0, +\infty)\).

Step 2: Draw the Graph of the Natural Logarithm

Start by graphing \(f(x) = \ln x\) as a reference, which is the parent function of the logarithmic functions. Remember that the graph of a logarithmic function always crosses the x-axis at 1, and the graph tends to infinity on the right and never crosses the y-axis on the left. Since \(\ln 1 = 0\), the graph passes through the point (1,0).

Step 3: Sketch the Graph of the Function

To sketch the graph of \(f(x) = \ln 2x\), note that the coefficient 2 in 2x implies a horizontal compression of the graph by a factor of 1/2. This means the graph of the function will be steeper than that of \(f(x) = \ln x\), but still with the same general shape, getting gradually steeper as x increases. Start at the x-axis at point (1,0) and draw the graph so it tends towards infinity as x increases. It should not cross the y-axis as \(x>0\) because the function is undefined at \(x=0\) or for \(x<0\).

Key Concepts

Domain of a function

When working with logarithmic functions, understanding the domain is crucial. The domain of a function is the set of all possible input values (x-values) for which the function is defined. For the natural logarithm function , the argument (in this case, the expression "2x") must be greater than zero. This is because logarithms of non-positive numbers are undefined in the set of real numbers.

In the exercise, the function given is \(f(x) = \ln(2x)\). To determine its domain, we need the argument of the logarithm, \(2x\), to be greater than zero. This translates to the inequality \(2x > 0\). Solving for \(x\), we divide both sides by 2, yielding \(x > 0\).

Thus, the domain of \(f(x) = \ln(2x)\) is all real numbers greater than zero, written as \(x \in (0, +\infty)\). This means the function is defined for any positive value of \(x\) and is undefined for \(x \leq 0\). Understanding this helps in avoiding mistakes when solving mathematical problems involving logarithms.

Graphing techniques

Graphing techniques allow for a visual representation of mathematical functions, offering insights into their behavior. Logarithmic functions, such as \(f(x) = \ln(x)\), have specific characteristics that are best understood through graphing.

When sketching the function \(f(x) = \ln(2x)\), it's beneficial to start with the basic parent function \(f(x) = \ln(x)\). This function typically features:
- An intercept at (1,0), since \(\ln(1) = 0\).
- A curve that rises to infinity as \(x\) increases.
- A trend towards negative infinity as \(x\) approaches zero from the right, without ever crossing the y-axis.

For \(f(x) = \ln(2x)\), the extra factor of 2 causes a horizontal compression. This compression makes the graph appear steeper compared to the graph of \(\ln(x)\). The point (1,0) serves as a starting position, but due to the compression, you'll notice that the function looks like it contracts towards the y-axis, but never intersects it, confirming \(x > 0\). Visualizing compression and other transformations improves our understanding of how logarithm functions behave.

Natural logarithm

The natural logarithm is a fundamental concept in mathematics, commonly symbolized as \(\ln(x)\). This logarithm is "natural" because it is based on the constant \(e\), an irrational number approximately equal to 2.718.

The natural logarithm function \(f(x) = \ln(x)\) has several key properties:
- \(\ln(1) = 0\), since \(e^0 = 1\), which provides a basis for its x-intercept at the point (1,0).
- It is undefined for \(x \leq 0\), which naturally impacts the domain of the function.
- As \(x\) approaches zero from the positive side, \(\ln(x)\) plunges towards negative infinity.
- The function rises gradually, becoming slower as \(x\) increases.

Applications of the natural logarithm extend beyond pure mathematics into fields like physics and finance, where growth processes are naturally modeled by exponential functions and their inverses, the logarithms. Mastery of the natural logarithm can be powerful in analyzing exponential change.