In words, we can say that the logarithm of x with base a is y.
A logarithm tells us the "number of times" a base is multiplied to get a specific number.
For example, tells us a number of times 2 is multiplied to get the answer 4. Similarly, tells us the number of times 3 is multiplied by itself to yield the number 9.
Just like addition and subtraction are two opposite operations, logarithms and exponential functions are opposite to each other. A logarithmic function can be converted into an exponential form like this:
In words, we can say that a raised to the power y is x.
We know that when any number is raised to a power 0, the answer is 1. In logarithmic form, we can conclude that the logarithm of a number 1 is equal to 0. Mathematically it can be written as .
Remember that the base of the exponential and logarithmic function is the same. For example, the logarithmic form of the exponential function is . The base of the exponential function is , whereas the base of the logarithmic function is also . Remembering this relationship will be very useful for you in understanding the logarithmic functions and will help to avoid any confusion while converting them into the exponential form or vice versa.
In this article, we will see how learn graphing logarithms, evaluating simple logarithms, natural and common logarithms and applying logarithm rules to solve complex logarithmic functions.
If we take the inverse of any function and graph it on a coordinate axis, then the graph of the inverse will be a reflected form of the graph of the original function. Since logarithmic functions are inverse of exponential functions, therefore the graph of the logarithmic function will be a reflection of the graph of the exponential function. Lets us consider an exponential function . Graphically, it will be represented as follows:
The logarithmic form of the function is . The graph of the logarithmic function will be reflected about the line .
It is pertinent to note that the graph of an exponential function is passing through the points (0,1) whereas the graph of a logarithmic function is passing through the points (1,0). The values in the exponential function become the values in the corresponding logarithmic function and values in the exponential function become values in the corresponding logarithmic function. This makes construction of a table of values easier while graphing logarithmic functions because you can simply reverse the values of corresponding exponential functions to get the tables of values for the logarithmic function.