Problem 104 - Introductory and Intermediate Algebra for College Students 4th

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

Answer

The relationship among the graphs is that 'y= log(10x)' is a horizontal compression by a factor of 1/10 of the function 'y= log(x)', while the graph of 'y=log(0.1x)' is a horizontal stretch by a factor of 10 of the function 'y= log(x)'. This is explained by the change of base property in logarithms.

Step 1: Graphing the functions

The first task is to graph the given functions. This requires a brief revision of the logarithmic function characteristics. The parent function, 'y= log (x)', has a vertical asymptote at x=0 and passes through the point (1, 0). Then, using a graphing tool, graph the three functions 'y=log(x)', 'y=log(10x)', and 'y=log(0.1x)'.

Step 2: Describe the transformations

Next, observe the transformations from the parent function to the other two functions. The graph of the function 'y= log(10x)' is a horizontal compression by a factor of 1/10 of the parent function 'y= log(x)', while the graph of the function 'y=log(0.1x)' is a horizontal stretch by a factor of 10 of the parent function 'y= log(x).'

Step 3: Identify the logarithmic property

The property of logarithms that explains this relationship is the change of base property. This property states that for any positive numbers a, b, and c, where a != 1 and b != 1, the equation log_a(b) = log_c(b) / log_c(a) holds. In this case, both 10 and 0.1 are multiples of the original base (which is 10 in basic logarithms), explaining the horizontal compressions and stretches observed in the graphs.