Problem 834 Construct the graph of \(\mathrm... [FREE SOLUTION]

Chapter 26: Problem 834

Construct the graph of \(\mathrm{y}=\log _{2} \mathrm{x}\).

Short Answer

Expert verified

To construct the graph of \(y = \log_2{x}\), first plot some key points: (1,0), (2,1), and (0.5,-1). Then, connect these points with a smooth curve that approaches the y-axis but never touches it, keeps rising as x increases, and crosses the x-axis at the point (1,0). The resulting curve is the graph of \(y = \log_2{x}\).

Step by step solution

Understanding the logarithm function

A logarithm function is denoted as \(\log_{b}{x}\), where b is the base of the logarithm and x is the value we are taking the logarithm of. It is the inverse function of an exponential function, which means that if y = \(\log_{b}{x}\), then x = \(b^{y}\). The logarithm function will satisfy the following conditions: 1. The domain is all positive real numbers (x > 0) 2. The range is all real numbers, i.e., (-鈭�, +鈭�) 3. If y = \(\log_{b}{x}\), then x = \(b^{y}\) 4. \(\log_{b}{1} = 0\), which means the graph always passes through the point (1,0). We will use these properties to plot the graph of y = \(\log_{2}{x}\).

Plot key points of the logarithm function

We will now plot some key points of the logarithm function using our knowledge of logarithms and the corresponding exponential functions: 1. When x = 1, y = \(\log_{2}{1} = 0\). This gives us the point (1,0). 2. When y = 0, x = \(2^{0} = 1\). This gives us the point (1,0) again. 3. When y = 1, x = \(2^{1} = 2\). This gives us the point (2,1). 4. When y = -1, x = \(2^{-1} = 0.5\). This gives us the point (0.5,-1). These key points will help us sketch the graph of the logarithm function.

Sketch the curve of the logarithm function

Now that we have some key points, we will connect these points with a smooth curve to get the graph of y = \(\log_{2}{x}\). The graph will resemble the following characteristics: 1. It will pass through the points (1,0), (0.5,-1), and (2,1). 2. The curve will approach the y-axis but never touch or cross it, as there is no value of y for x = 0 in the logarithm function. 3. The curve will continue to rise as x increases, approaching positive infinity. 4. The curve will cross the x-axis at the point (1,0), as \(\log_{b}{1} = 0\). After drawing the curve with the characteristics mentioned above, we will have the graph of y = \(\log_{2}{x}\).

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91影视

Construct the graph of \(\mathrm{y}=\log _{2} \mathrm{x}\).

Short Answer

Step by step solution

Understanding the logarithm function

Plot key points of the logarithm function

Sketch the curve of the logarithm function

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