Question

Describe three ways to transform the graph of \(f(x)=\log x\) to obtain the graph of \(g(x)=\log 100 x-1\). Justify your answers.

Short answer

The two transformations required to obtain the graph of \(g(x) = \log 100x - 1\) from \(f(x) = \log x\) are: 1) A horizontal compression of the graph towards the y-axis by a factor of 100; 2) A vertical shift downwards by 1 unit.

Step-by-step solution

Recognizing the form of the function

In the function \(g(x) = \log 100x -1\) we can break it down into two transformations of the original function \(f(x) = \log x\). First, there is a multiplication inside the logarithm, which will stretch the graph horizontally. Second, there is a subtraction after the logarithm, which will shift the graph downwards.

Describing the transformations

The first transformation factor ‘100’ inside the logarithm is a horizontal compression because any value greater than 1 inside the logarithm compresses the graph. In simple terms, the graph of \(f(x) = \log x\) will be compressed towards the y-axis by a factor of 100 to obtain the graph \(f(x) = \log 100x\).The second transformation is a vertical shift. This is due to the ‘-1’ after the logarithm which shifts the graph downwards by 1 unit. That is, every point on the graph of \(f(x) = \log 100x\) has been shifted 1 unit downwards along y-axis to derive the graph \(g(x) = \log 100x −1\).

Key concepts

Horizontal Compression

Understanding how a horizontal compression affects a graph is essential in transforming logarithmic functions. This concept applies when a factor inside the logarithmic function is greater than 1, as seen with the function \(g(x) = \text{log} 100x\). Imagine the x-axis as a rubber band. If we compress the band by a factor of 100, all points along the axis move closer to the y-axis. In mathematical terms, every x-value in the original graph of the function \(f(x) = \text{log} x\) would be reduced by 100 times to fit the new scale. This intensity of scaling results in a graph that appears steeper compared to the original. It's important not to confuse this with a stretch—a horizontal compression effectively 'squeezes' the graph horizontally without changing its vertical extent.

To provide additional clarity on this topic, let's consider the graph of \(f(x) = \text{log} x\). Without any alterations, this graph would pass through points such as \( (10, 1) \) and \( (100, 2) \). However, after a horizontal compression by a factor of 100, these points would transform to \( (0.1, 1) \) and \( (1, 2) \) respectively, thus bringing the curve closer to the y-axis and increasing its steepness.

Vertical Shift

The concept of a vertical shift is easier to visualize and grasp than many other transformations. When we talk about a vertical shift in the context of logarithmic functions, such as the transition from \(f(x) = \text{log} 100x\) to \(g(x) = \text{log} 100x - 1\), you can think of it as moving the entire graph up or down along the y-axis. In this case, the '-1' indicates a downward shift of 1 unit.

Every coordinate point \( (x, y) \) on the original graph will have its y-value decreased by 1 to become \( (x, y-1) \). To further exemplify, if the original graph had a point \( (100, 2) \) after \(f(x)\), this point would now be at \( (100, 1) \) on the transformed graph \(g(x)\). This shift does not alter the shape or orientation of the graph but merely changes its position in relation to the horizontal axis. It's a practical and straightforward transformation that can significantly alter the equations and inequalities involving the given function.

Graph of Logarithmic Functions

When studying the graph of logarithmic functions, we focus on how they typically progress: slowly increasing as they move from left to right, never touching the negative y-axis due to their undefined nature at zero and negative x-values. The basic form of a logarithmic function graph is \(f(x) = \text{log} x\), which passes through the point \( (1, 0) \)—since the log of 1 is always 0—and approaches infinity as x approaches infinity.

The curve is relatively flat as it moves away from the y-axis, but it becomes steep very close to it. This characteristic shape is why even slight transformations, such as horizontal compression or vertical shift, can have a visually significant impact on the graph. Furthermore, understanding these transformations is crucial in real-world applications, such as sound decibels and earthquake magnitudes, which both use logarithmic scales. By mastering the ability to graph basic logarithmic functions and their transformations, students can gain deeper insight into a variety of natural phenomena and mathematical problems.