Why is log(base 1/a)x a reflection over the y axis?
To understand why log(base 1/a)x is a reflection over the y-axis, we need to understand the properties of logarithmic functions and their graphs.
The general logarithmic function is defined as y = log(base a)x, where "a" is the base and "x" is the argument. The logarithmic function represents the inverse of the exponential function y = a^x.
Now, let's discuss the specific case of log(base 1/a)x:
1. The base: In this case, the base of the logarithmic function is 1/a. If we consider a positive value of "a" (greater than 1), the base 1/a is between 0 and 1, indicating a fraction. For example, if a = 2, then the base 1/a is 1/2, which lies between 0 and 1.
2. Reflection over the y-axis: When a logarithmic function has a base between 0 and 1, such as 1/a, the graph of the logarithmic function will be reflected over the y-axis compared to the graph of a logarithmic function with a base greater than 1. This means that the points on the graph will appear on the opposite side of the y-axis.
The reason for this reflection is that the base determines the behavior of the function. A base greater than 1 results in a logarithmic function that increases as x increases, whereas a base between 0 and 1 results in a logarithmic function that decreases as x increases. Therefore, reflecting the graph of the logarithmic function with a base between 0 and 1 over the y-axis reveals this decreasing behavior.
So, log(base 1/a)x is a reflection over the y-axis because the base 1/a is between 0 and 1, causing the graph of the logarithmic function to be reflected compared to a logarithmic function with a base greater than 1.