How does the equation of a log and Ln function effect the graph. (Ex. Dilation, reflection, ect.)
The equation of a logarithmic function and a natural logarithmic function (ln) affects the graph in several ways, including dilation, reflection, and translation. Let's break it down:
1. Dilation: Changing the base of a logarithmic function (log) affects the vertical dilation of the graph. The base determines how steep or flat the graph appears. For example, if the base is greater than 1, the graph is compressed vertically, making it steeper. Conversely, if the base is between 0 and 1, the graph is stretched vertically, becoming flatter.
2. Reflection: The graph of a logarithmic function can be reflected over the x-axis by adding a negative sign before the function, such as -log(x). This reflection causes the graph to switch from being increasing to decreasing, or vice versa. Reflection allows for transformations of the graph across the x-axis.
3. Translation: Similar to other function graphs, the logarithmic function can be translated horizontally or vertically. For example, adding or subtracting a constant term inside the function, like log(x + a) or log(x - a), results in a horizontal shift of the graph to the left or right, respectively. Meanwhile, adding or subtracting a constant term outside the function, such as log(x) + b or log(x) - b, causes a vertical shift up or down correspondingly.
Similarly, the natural logarithmic function (ln) follows the same transformation rules as the logarithmic function. It can be dilated, reflected, and translated in a similar manner.
To summarize, changing the base of a logarithmic function (log) affects the vertical dilation, adding a negative sign allows for reflection over the x-axis, and adding/subtracting constants inside or outside the logarithmic function enables horizontal or vertical translations. The same rules apply to the natural logarithmic function (ln).