Problem 114 - College Algebra

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 between these graphs is explained by the logarithmic relationship where the multiplication inside a logarithm turns into addition, shifting the graph left or right. When the coefficient of \(x\) is multiplied by 10, the graph for \(y = \log (10x)\) shifts left, and when divided by 10, the graph for \(y = \log (0.1x)\) shifts right.

Step 1: Graphing the First Logarithm

Start by graphing \(y = \log x\), the simplest logarithmic function. The graph will start from (1,0) and increase as \(x\) moves away from 1, demonstrating the property of logarithms where \(\log_b a = n\) only if \(b^n = a\). Thus for base 10, when \(x = 1\), \(y = 0\) because \(10^0 = 1\).

Step 2: Graphing the Second Logarithm

Next, graph the function \(y = \log (10x)\). This graph will show a shift to the left compared to \(y = \log x\), because the coefficient of \(x\) is greater than 1, effectively dividing the \(x\)-values by 10. This demonstrates one of the key properties of logarithms, that \(\log_b (ac) = \log_b a + \log_b c\). When \(x = 0.1\), \(y = 0\), because \(10 \times 0.1 = 1\) and \(\log 1 = 0\).

Step 3: Graphing the Third Logarithm

Finally, graph \(y = \log (0.1x)\). With a smaller \(x\) coefficient, this graph shifts to the right of \(y = \log x\), effectively multiplying the \(x\)-values by 10 as compared to \(y = \log x\). When \(x = 10\), \(y = 0\) since \(0.1 \times 10 = 1\) and \(\log 1 = 0\).

Step 4: Analyzing the Relationship

Analyzing these three graphs, it's clear that changing the coefficient of \(x\) in a logarithm shifts the graph horizontally. Greater coefficients shift the graph towards the left, and smaller ones towards the right. As the coefficient multiplies or divides \(x\), \(\log (ax)\) can be written as \(\log a + \log x\), according to properties of logarithms. This relation results in the shifting of these graphs.