 Hello and welcome to the session. In this session, we will discuss log given of a number. Now let a, y and b are three numbers such that a raised to power y is equal to b. Then, y is said to be the log given b to the base. It is written as log b to the base a is equal to y. That is, when a raised to power y is equal to b, then y which is the power is called the log b to the base a. Now let us discuss this with the help of an example. Now we have 10 raised to power x is equal to 100 which implies 10 raised to power x is equal to 10 raised to power. Now here this is the same so we can equate the powers. So this implies x is equal to, now put in this value in this equation. We have 10 raised to power 2 is equal to 100. Now in the log given form, we can write it as log 100 to the base 10 is equal to 2. Therefore, log of a number that is log of 100 to the base 10 is equal to 2 and here 2 10 is raised. Just one more example. Here 2 raised to power x is equal to a, 2 raised to power x is equal to 2, 2 raised to power 3 which implies x is equal to 3. This equation as x is equal to 3 equals u. That is, we are writing the power is equal to 3. That means this power which is 3 is equal to log 8 to the base 2. Now let us discuss one more example. Here 2 raised to power x is equal to 5 between if the base 2 is equal to 2 and 3. For example such as clear that log of the number of a given base should be raised to get the given number. 2 raised to power 0 is equal to 1. So let us take a raised to power 0 is equal to 1. Now this implies in log given form it can be written as log 1 to the base a is equal to 0. Now in log given form it can be written as log a to the base a is equal to 1. So these are two very important results which are to be remembered. And from this result the log given as 1 to every base is 0 to the number. To the same base this few results which are real numbers to the base a is equal to x. Let log x to the base a is equal to z, z is equal to x. Now this implies in this equation it will be a raised to power log x to the base a is equal to x. We have proved this result. Now let us discuss the second result which is for a is greater than 0 equal to 1 is a is equal to log y to the base a. Now let us prove it that log x to the base a is equal to p and log y to the base a is equal to q. And where x y greater than 0 and a is greater than 0 and also a is not equal to 1. Now by definition of logarithm we can write this equation as a raised to power p is equal to x and this equation as equal to y equal to a raised to power q. So this implies this is of b. This implies log x to the base a is equal to log y to the base a. So we have proved this result. This a is greater than log y to the base a. Now let us prove this result. Now using this result we can write x is equal to a raised to power log x to the base a. Here log x to the base a is greater than and here we can write y as a log y to the base a. Now the bases are same so we can compare the parts. So log x to the base a is greater than log y to the base a. That is an increasing function. Now let us discuss one example. Then three. This implies log base 10 is greater than log 3 to the base 10. Now here a is greater than 1. That is a is 10 here which is greater than 1. This implies log x to the base a that is log 5 to the base 10 is greater than log y to the base a. That is log 3 to the base 10. Now let us discuss the next result which is if 0 is less than a is less than 1. That is a is lying between 0 and 1. And log x to the base a is less than log y to the base a. Let us prove this result. But log x to the base a is greater than a raised to power log y to the base a. Log x to the base a is less than log y to the base a. That is log x to the base a is a decreasing function. Result with the help of an example. Now here log is equal to 1 by 2. That is a is lying between 0 and 1. And x is greater than y. That is 16 log 16 to the base 1 by 2 is equal to minus 4. And log a to the base 1 by 2 is equal to minus 3 is less than minus 3. Log 16 to the base 1 by 2 is less than log a to the base 1 by 2. Therefore if a is lying between 0 and 1 and x is greater than y. Log x to the base a log y to the base log rhythmic functions. Exponential functions. The graph of the logarithmic function is the mirror image of the graph of the exponential function above the line y is equal to x. Let us discuss graph of logarithmic function. These are the graphs of the logarithmic functions. That is y is equal to log x to the base 2. Then for y is equal to log x to the base 3. First of all let us draw a graph for y is equal to log x to the base 2. Now this implies in exponential form we can write it as draw a table for the different values of x and y. Now for y is equal to 0. That is putting y is equal to 0 in this equation we get 0 which is 1. So for y is equal to 0 x is equal to 1. Then for y is equal to 2 raised to power 2 which is 4. Now let us we have plotted all these points on the graph. Now by joining all these points we have written the graph for the logarithmic function. y is equal to x to the base equal to log x to the base 3. Base 3 implies 3 raised to power 1. Now we have drawn the table for the different values of x and y. Now for y is equal to 0 1 is equal to 2 power 2 which is 9. And now let us plot these points on the graph. So we have plotted these 3 points on the graph. Now by joining all these points we are getting the graph for the logarithmic function. y is equal to log x to the base 3. Now let us draw a graph y is equal to log x to the base 4. It implies 4 raised to power y. Now for y is equal to we get x is equal to 1. Then for y is equal to 1 we get x is equal to 4. And for y is equal to minus 1 x is equal to 4 raised to power minus 1 which is equal to 1 by 4. And this is equal to 0.25. These points on the graph we have plotted these 3 points on the graph. Now by joining all these points we are getting the graph of the logarithmic function. y is equal to log x to the base 4. Now here you can observe that curves that are the graphs of the logarithmic functions are passing through the common point that is 1, 0. That is these graphs are meeting their x-axis at the point 1, 0. In this session we have learnt about logarithmic number and logarithmic function. So this completes our session. Hope you all have enjoyed the session.