 Hello and welcome to the session. In this session we will learn about logarithmic function. Now we know that the function f which sets up a one to one correspondence between its domain a and its range v has an inverse that is the inverse function which is denoted by f inverse from v to a and y is equal to f of x if I only x is equal to f inverse of y that is the inverse of a function exists only when for every value of x that is the domain we will get a unique value of y that is the range and if we get the same y on two different values of x then inverse will not exist. Now consider this graph this is the graph of y is equal to a is to power x that is the exponential function x is equal to a is to power x is the exponential function where a is greater than 1. Now here you can see that for every real number x there exists positive real number y similarly for every positive real number y there exists a unique real number x satisfying the function that is this function which is equal to a is to power x where a is a fixed number so the function f of x sets up a one to one correspondence between the set of real numbers on x axis that is the domain the set of positive real numbers on o y that is the range so the function f of x which is equal to a is to power x has an inverse function it's a one to one correspondence between its domain and range so it has an inverse function which is called the log mythic function to the base a denoted by log y to the base a so y is equal to a is to power x implies x is equal to log y to the base a. Now let us see the graph of this logarithmic function now this curve is the graph of the logarithmic function y is equal to log x to the base a which is the real image of y is equal to a is to power x in the line y is equal to x. Now let us discuss the definition of logarithm of a number now if three numbers a x and y are so related that a is to power x is equal to y then x is set to be the logarithm of a number y to the base a it is represented log y to the base a is equal to x. Now let us discuss two special logarithms now y is equal to a is to power x implies log y to the base a is equal to x now in the first case 1 is equal to a is to power 0 that is anything is to power 0 is equal to 1 so we have taken 1 is equal to a is to power 0 and in the logarithmic form we can write it as by using this result log 1 to the base a is equal to 0 that is log 1 to any base is equal to 0 and secondly a can be written as a is to power 1 which implies log a to the base a is equal to 1 that is log of a number to that base is equal to 1. Now for y is equal to log x to the base a if x is equal to 1 then y is equal to log 1 to the base a which is equal to 0 by using this result which we have discussed earlier so we obtain a point whose coordinates are 1 0 on the log of the logarithmic function and if x is equal to a then y is equal to log a to the base a which by using this result will be equal to 1 thus we obtain a point whose coordinates are a 1 on the graph. Now the domain of the logarithmic functions is the set all positive real numbers and the range is the set r of all real numbers. Now let us discuss the graph of log x to the base a where a and x are any positive numbers. Now let us discuss the case 1 when a is greater than 1 and the logarithmic function is y is equal to log x to the base a or you can say x is equal to a raised to power y. Now let us discuss this case with the help of an example and here we have to draw a graph of the logarithmic function y is equal to log x to the base 3. Now here a is equal to 3 which is greater than 1. Now y is equal to log x to the base 3 or you can write x is equal to 3 raised to power y. First of all let us draw a table for the different values of x and y. Now the function is given as x is equal to 3 raised to power y. Now for y is equal to 1, x is equal to 3 raised to power 1 which is equal to 3. So for y is equal to 1, x is equal to 3 and for y is equal to 2, x is equal to 3 raised to power 2 which is equal to 9. So for y is equal to 2, x is equal to 9. Now for for y is equal to 0, x is equal to 3 raised to power 0 which is equal to 1. So for y is equal to 0, x is equal to 1. And for y is equal to minus 1, x is equal to 3 raised to power minus 1 which is equal to 1 by 3 which is equal to 0.3. So for y is equal to minus 1, x is equal to 0.3. Now for y is equal to minus 2, x is equal to 3 raised to power minus 2 which is equal to 1 over 3 raised to power 2 which is equal to 1 over 9 which is equal to 0.1. So for y is equal to minus 2, x is equal to 0.1. Now let us plot these points on the graph. Now the first point that we will plot on the graph will be 3, 1. So this is the required point on the graph. And now we will plot the point 9, 2 on the graph. Similarly, we will plot all the other points on the graph. So we have plotted all the points on the graph. So this is the graph of the logarithmic function y is equal to log x to the base 3 or x is equal to 3 raised to the power y. Now you can see that the curve does not pass through the origin and it cuts the x-axis when y is equal to 0 i.e. when x is equal to 1 i.e. the curve meets the x-axis at the point 1, 0 i.e. this point. And this curve cuts the y-axis when x is equal to 0 i.e. when y is equal to minus infinity. Therefore the curve meets the negative direction of y-axis at infinity. So as x tends to 0 i.e. on this side y tends to minus infinity. And x is positive whether y is positive or negative or in other words this curve lies only in the first and the fourth quadrant i.e. through the right of y-axis. And from x is equal to 3 raised to the power y we get that x increases as y increases when a which is 3 here is greater than 1. And on this side you can see that as x tends to infinity, y tends to infinity. Now you can also observe that as x increases from 0 to plus infinity, y increases from minus infinity to plus infinity. Therefore the domain of a logarithmic function is a set of all positive real numbers and the range is the set of all real numbers. Now let us discuss the case 2 when 0 is less than a less than 1, y is equal to log x to the base a or x is equal to a raised to power y. Now let us discuss this with the help of an example. Now here draw a graph of y is equal to log x to the base 1 by 3 or x is equal to 1 by 3 whole raised to the power y. Now here a is equal to 1 by 3 which is lying between 0 and 1. Now for plotting a graph for the given function let us draw a table for the different values of x and y. Now here given x is equal to 1 by 3 whole raised to the power y so far y is equal to 0, x is equal to 1 by 3 whole raised to the power 0 which is equal to 1. So when y is equal to 0 x is equal to 1. For y is equal to 1, x is equal to 1 by 3 whole raised to the power 1 which is equal to 1 by 3 which is equal to 0.3 to 2, x is equal to 1 by 3 whole raise to power 2 which is equal to 1 by 9 which is equal to 0.1. So for y is equal to 2, x is equal to 0.1. Now for y is equal to minus 1, x is equal to 1 by 3 whole raise to power minus 1 which is equal to 3 raise to power 1 which is equal to 3. So for y is equal to minus 1, x is equal to 3, and for y is equal to minus 2, x is equal to 1 by 3 whole raise to power minus 2 which is equal to 3 raise to power 2 which is equal to 9. So for y is equal to minus 2 x is equal to 9 and now we will plot all these points on the graph. First of all let us plot the point 1 0 on the graph. So this is the point 1 0 on the graph. Now we will plot the point 0.3 1 on the graph. So this is the required point on the graph. Similarly we will plot all these points on the graph. So we have plotted all the points on the graph. Now by joining all these points we are getting the graph that is the graph for the given log rhythmic function which is y is equal to log x to the base 1 by 3 or x is equal to 1 by 3 whole raise to power y. Now here the first point that you can observe is that the curve does not pass through the origin O and secondly the curve cuts the x axis when y is equal to 0 and x is equal to 1 that is the curve the x axis at the point 1 0 as the curve cuts the y axis when x is equal to 0 that is when y is equal to infinity. Therefore the curve reach the positive direction of y axis at infinity that is as x tends to 0 y tends to infinity and x is positive that is y is positive or negative or in other words the curve lies only in the first and the fourth quadrant that is to the right of y axis and from x is equal to 1 by 3 whole raise to power y we get that x increases as y decreases that is when the value of a lies between 0 and 1. So on this side you can observe that as x tends to infinity y tends to minus infinity. So this was the second case when a is lying between 0 and 1 for the log rhythmic function y is equal to log x to the base 1 by 3. So in this session you have learnt about log rhythmic functions and this completes our session hope you all have enjoyed the session.