Problem 114 - Algebra and Trigonometry

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

All the graphs intersect at (1,0) and have undefined regions for negative x values. The graph of \(y=\log(10x)\) is a compressed version of \(y=\log(x)\), and the graph of \(y=\log(0.1x)\) is an expanded version of \(y=\log(x)\). This relationship is due to the scale-change property of logarithms.

Step 1: Graph y=log(x)

Begin by graphing the function \(y=\log(x)\). Remember the common characteristics of logarithmic graphs: An undefined negative x region due to the log function's asymptote, an intersect at (1,0) and a general shape that gradually increases as x goes to positive infinity. Draw this using these characteristics.

Step 2: Graph y=log(10x)

Next, graph the function \(y=\log(10x)\). The 10x essentially causes the graph to compress horizontally by a factor of 10, which means the graph appears narrow. This effect is due to the 'scale-change' property of logarithms, which states log_b(a*cx) = log_b(a) + log_b(c), where b is the base, c is the multiplier, and a is the variable. Draw the graph according to these specifications and observe the changes.

Step 3: Graph y=log(0.1x)

Finally, graph the function \(y=\log(0.1x)\). The '0.1x' expands the graph horizontally by a factor of 10, effectively moving it 10 times further from the y-axis compared to the graph of \(y=\log(x)\). It should be noted this function exhibits reverse behavior from \(y=\log(10x)\) because of the 0.1 in front of the x-variable. Draw the graph according to this interpretation.

Step 4: Interpret the relationships

Once all three graphs are sketched, the relationships can be visually interpreted. This involves comparing how shifting the original function horizontally by factors of 10 (scaling by factors of 1/10 or 10) either compressed or expanded the graph (for \(y=\log(10x)\) or \(y=\log(0.1x)\) respectively). Note that these transformations show directly the effect of a scale-change (log_b(a*cx) = log_b(a)+log_b(c)) in logarithmic functions.