Problem 114

Question

Graph $y=\log x, y=\log (10 x),$ and $y=\log (0.1 x)$ in the same viewing rectangle. Describe the relationship among the three graphs. What logarithmic property accounts for this relationship?

Step-by-Step Solution

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Answer

Graphs of the functions $y = \log x$, $y = \log(10x)$ and $y = \log(0.1x)$ on the same viewing rectangle show that multiplying the argument of the logarithm by a factor results in horizontal compression or stretching of the base graph $y = \log x$. This behavior arises due to the logarithmic property: $\log_b(ax) = \log_b a + \log_b x$.

1Step 1: Graph the First Function

Begin by graphing the function $y=\log x$. Recall the basic shape of a logarithmic function. The graph of a logarithm function $y = \log_b x$ is a curve which increases from left to right, passing through the point ($1, 0$), and has the x-axis (the line $y = 0$) as a horizontal asymptote. For the function $y=\log x$, the base 'b' is 10 (default base for logarithm).

2Step 2: Graph the Second Function

Next, graph the second function $y=\log(10x)$. Multiplying the argument of the logarithm by a scalar (10 in this case) results in a horizontal compression of the graph. The graph of $y=\log(10x)$ will be 'narrower' compared to the graph of $y=\log x$.

3Step 3: Graph the Third Function

Lastly, graph the third function $y=\log(0.1x)$. Multiplying the argument of the logarithm by 0.1 will result in a horizontal stretch of the graph. The graph of $y=\log(0.1x)$ will be 'wider' compared to the graph of $y=\log x$.

4Step 4: Observe and Describe the Relationship

Observe the three graphs together. Notice the horizontal shifts caused by the multiplication of the logarithmic argument by values other than 1. In logarithmic form, there is a property that states that $\log_b(ax) = \log_b a + \log_b x$. With this property in mind, graph transformations can be better understood. It is evident that multiplying the argument of the logarithm by 10 causes the graph to compress, while multiplying it by 0.1 causes it to stretch.

Problem 114

Problem 115

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