Question

Without using a graphing utility, sketch the graph of \(y=\log _{2} x .\) Then on the same set of axes, sketch the graphs of \(y=\log _{2}(x-1), y=\log _{2} x^{2}\) \(y=\left(\log _{2} x\right)^{2},\) and \(y=\log _{2} x+1\)

Short answer

Based on the step-by-step process of sketching the given logarithm functions, provide a short-answer explanation of the transformations that occur for each function: To sketch the graph for each given function, we apply specific transformations to the basic logarithm function \(y=\log_{2}x\): 1. \(y=\log_{2}(x-1)\): Apply a horizontal shift of 1 unit to the right. The vertical asymptote moves to \(x=1\) and the curve passes through \((2, 0)\). 2. \(y=\log_{2}x^2\): Replace \(x\) with \(x^2\), causing the graph to reflect about the y-axis, and becoming symmetric with respect to the y-axis. The curve still passes through \((1, 0)\). 3. \(y=(\log_{2}x)^2\): Square the basic logarithm function, which bends the curve upward with a gentle slope near the y-axis. The graph still passes through \((1, 0)\). 4. \(y=\log_{2}x+1\): Apply a vertical shift of 1 unit upwards. The vertical asymptote remains at \(x=0\) but the graph now passes through \((1, 1)\).

Step-by-step solution

Introduction to logarithms

A logarithm is the inverse function of exponentiation. In the given exercise, we will work with logarithms with a base of 2, which is represented as \(\log_{2}x\). The basic graph of \(y=\log_{2}x\) has the following properties: - The domain is \((0, \infty)\) - The range is \((-\infty, \infty)\) - The graph is always increasing - The graph has a vertical asymptote at \(x=0\), - The graph passes through the point \((1, 0)\).

Sketching \(y=\log_{2}(x-1)\)

To sketch \(y=\log_{2}(x-1)\), we need to apply a horizontal shift to the basic logarithm function. This shift will move the graph one unit to the right. The new vertical asymptote will be at \(x=1\), and the curve will pass through the point \((2, 0)\).

Sketching \(y=\log_{2}x^2\)

To sketch \(y=\log_{2}x^2\), we need to replace \(x\) in the basic logarithm function with \(x^2\). This will cause the graph to be reflected about the y-axis, giving us a curve that is symmetric with respect to the y-axis. The graph will still pass through the point \((1, 0)\).

Sketching \(y=(\log_{2}x)^2\)

To sketch \(y=(\log_{2}x)^2\), we need to square the basic logarithm function, which will make the curve to bend upward. The shape of the graph will still pass through the point \((1, 0)\), but the slope will be gentle near the y-axis.

Sketching \(y=\log_{2}x+1\)

To sketch \(y=\log_{2}x+1\), we need to apply a vertical shift to the basic logarithm function. This shift will move the graph one unit upwards. The vertical asymptote will still be at \(x=0\), but the curve will now pass through the point \((1, 1)\). Now, having all the individual graphs of the functions above, we can sketch them on the same set of axes. This will enable us to compare their properties and analyze their differences visually.