Problem 15 - College Algebra and Calculus: An Applied Approach

Question

In Exercises, sketch the graph of the function. $$ y=\ln 2 x $$

Step-by-Step Solution

Answer

The graph of the function \(y = \ln(2x)\) would be 'skinnier' than the graph of \(y = \ln(x)\), with an x-intercept at \(x=0.5\) and a vertical asymptote at \(x=0\).

Step 1: Understand the basic logarithmic function

The basic logarithmic function \(y = \ln(x)\) is increasing, defined for x > 0 and has an x-intercept at \(x=1\). It has a vertical asymptote at \(x=0\).

Step 2: Understand the transformation

The function \(y = \ln(2x)\) is equal to \(\ln(x)\) but scaled horizontally by a factor of 0.5. This means that all x-values are divided by factor of 2, causing the graph to appear 'skinnier' compared to the graph of \(\ln(x)\). The x-intercept will shift to \(x=0.5\), but the vertical asymptote will remain at \(x=0\).

Step 3: Sketch the graph

Start by sketching the basic function \(y = \ln(x)\), mark the x-intercept at \(x=1\), and indicate the vertical asymptote at \(x=0\). Then sketch the function \(y = \ln(2x)\) by 'shrinking' the original graph horizontally by a factor of 0.5 so the x-intercept now lies at \(x=0.5\). Label your graph clearly to show these features.

Key Concepts

Logarithmic Scale Transformation

Understanding logarithmic scale transformation is crucial when dealing with functions like y = \(ln 2x\). When we talk about the transformation of a logarithmic function, it refers to changes in the function's appearance when plotted on a graph. These transformations are often a result of multiplying the input variable, x, by a constant or adding a constant to it.

In our case, the transformation involves scaling the input variable by a factor of \(2\). This scaling affects the spread of the function on the horizontal axis, essentially compressing or stretching it. For the function y = \(ln 2x\), the graph is horizontally compressed by a factor of \(0.5\) because every x value is effectively halved.

Imagine you're zooming in on the x-axis of a basic logarithmic function y = \(ln x\), making the curve look skinnier. That's the visual effect of logarithmic scale transformation—altering the rate at which the logarithmic curve ascends or descends along the axis.

Vertical Asymptote

A vertical asymptote is a vertical line that a graph approaches but never touches or crosses. It represents a boundary beyond which the function continues towards infinity or negative infinity. In logarithmic functions like y = \(ln x\) and y = \(ln 2x\), the vertical asymptote is always located where the input to the logarithm becomes zero.

For the function y = \(ln x\), this occurs at x = 0. No matter how the function is transformed, the asymptote remains fixed because the logarithmic function is only defined for positive values of x. Even after applying the scale transformation to y = \(ln 2x\), the vertical asymptote does not move because the logarithm of zero is undefined. This is a fundamental characteristic of logarithmic functions that dictates their behavior and overall shape on a graph.

X-Intercept

The x-intercept of a graph indicates the point where the curve crosses the x-axis. For logarithmic functions, determining the x-intercept involves finding the value of x that will make the function's output equal to zero.

Let's take the basic logarithmic function y = \(ln x\). The x-intercept here occurs at x = 1, because \(ln 1 = 0\). Any logarithmic function will cross the x-axis at the point where its input equals one since the logarithm of one is always zero, which is a key property of logarithms.

When we apply a horizontal scaling, like in y = \(ln 2x\), the x-intercept changes accordingly. In this case, the graph's x-intercept shifts to x = 0.5 because \(ln (2 \cdot 0.5) = ln 1 = 0\). Always remember, regardless of the transformation applied to the input, the point where the logarithmic function intersects the x-axis will shift in a way that maintains the property of logarithms—that the logarithm of one is zero.