Problem 66 - Algebra for College Students

Question

Graph \(f(x)=\ln x\). How should the graphs of \(f(x)=2\) \(\ln x, f(x)=4 \ln x\), and \(f(x)=6 \ln x\) compare to the graph of \(f(x)=\ln x\) ? Graph the three functions on the same set of axes with \(f(x)=\ln x\).

Step-by-Step Solution

Answer

The graphs of \(2\ln x\), \(4\ln x\), and \(6\ln x\) are steeper versions of \(\ln x\), with all intersecting at \((1, 0)\).

Step 1: Understand the Basic Function

The function \( f(x) = \ln x \) is the natural logarithm function. Its graph passes through the point \((1, 0)\) and increases slowly for \(x > 1\). As \(x\) approaches 0 from the right, the function approaches negative infinity.

Step 2: Analyze the Transformation

The functions \(f(x) = 2 \ln x\), \(f(x) = 4 \ln x\), and \(f(x) = 6 \ln x\) involve multiplying the natural logarithm by a constant, thus stretching the graph vertically. Each constant multiplies the output of \(\ln x\), making the curve steeper without altering \(x\)-intercepts.

Step 3: Graph the Basic Function

Plot the graph of \(f(x) = \ln x\). This curve should pass through \((1, 0)\) and be concave down as it moves rightward.

Step 4: Graph Transformed Functions

For \(f(x) = 2\ln x \), \( 4\ln x \), and \( 6\ln x \), plot each by creating steeper versions of \(\ln x\). Each should pass through \((1, 0)\). As constants increase, the graph steepens, but their trajectories follow parallel paths to the original \(\ln x\) graph.

Step 5: Compare the Graphs

On the same set of axes, observe that all graphs intersect at \((1, 0)\). \(2\ln x\), \(4\ln x\), and \(6\ln x\) are successively steeper, illustrating how multiplying by a constant vertically stretches the graph. All graphs remain concave down and approach negative infinity as \(x\) nears 0.

Key Concepts

Natural Logarithm Function

The natural logarithm function, denoted by \( f(x) = \ln x \), is a fundamental concept in mathematics. This function is only defined for positive values of \( x \), since the logarithm of zero or a negative number is undefined. The graph of \( \ln x \) is unique because it passes through the point \((1, 0)\). This means that the natural logarithm of one is zero. It provides a special point of reference in plotting the graph.

• For \( x > 1 \), the function increases slowly.
• As \( x \) approaches zero from the positive side, \( \ln x \) decreases without bound, approaching negative infinity.

Graphically, the function is concave down and will never touch or cross the y-axis. Its rate of increase diminishes as \( x \) grows larger.

Vertical Transformation

Vertical transformation in the context of logarithmic functions involves scaling the output of the function by a constant. For instance, multiplying \( \ln x \) by a constant like 2, 4, or 6 results in functions \( 2 \ln x \), \( 4 \ln x \), and \( 6 \ln x \). This alteration affects the graph by making it steeper.The multiplication stretches the graph vertically:

• The constant makes the curve rise more sharply as \( x \) increases.
• However, the \( x \)-intercept at \( (1, 0) \) stays the same since \( \ln 1 = 0 \) and any constant times zero remains zero.

This transformation does not affect the concavity or the direction of the graph, meaning it remains concave down. These functions become steeper as the constant increases, creating a more dramatic ascent.

Concave Down Behavior

The graph of the natural logarithm function \( \ln x \) and its transformed versions like \( 2 \ln x \), \( 4 \ln x \), and \( 6 \ln x \) are characterized by their concave down behavior. Concavity is about the shape of the graph: when a curve is concave down, it looks like an upside-down bowl or a frown.For \( \ln x \):

• The rate of increase diminishes as \( x \) gets larger.
• This results in a curve that continues to rise but at a slowing pace.

The concavity remains consistent even with vertical stretching. These graphs, no matter how steep, do not change their characteristic concave down nature. Hence, even with different multipliers, the steeper graphs retain the same frowning shape as the original \( \ln x \).

X-intercepts of Logarithmic Functions

Understanding the location of \( x \)-intercepts for logarithmic functions is crucial for accurate graphing. The \( x \)-intercept of a function is the point where the graph crosses the x-axis, equivalent to where the output of the function is zero.For the function \( f(x) = \ln x \):

• The \( x \)-intercept is \((1, 0)\) because \( \ln 1 = 0 \).
• The transformed functions \( f(x) = 2 \ln x \), \( f(x) = 4 \ln x \), and \( f(x) = 6 \ln x \) retain this intercept. This is due to the fact that multiplying zero by any constant remains zero.

Thus, all these logarithmic functions will intersect the x-axis at the same point \( (1, 0) \). This consistency aids in accurately sketching a graph and analyzing the behavior of logarithmic functions as they are modified by different transformations.