 Hello and welcome to the session. In this session we will discuss the inverse relationship between exponential and logarithmic functions and we will use this relationship to solve problems involving logarithms and exponents. Let us recall the definitions of exponential function and logarithmic function. An exponential function is of the form b raised to power y is equal to x such that b is greater than 0 and b is not equal to 1 and a logarithmic function is of the form y is equal to log of x to the base b where x and b are positive real numbers. Base b is not equal to 1. Also we know that the expression y is equal to log of x to the base b is equivalent to b raised to power y is equal to x. The definition of logarithmic function suggests a close relationship with an exponential function of the same base. In fact a logarithmic function is the inverse of the corresponding exponential function. Suppose we have to find inverse of an exponential function f of x is equal to b raised to power x. Now we replace f of x with y and we get y is equal to b raised to power x. Now we switch x and y and we get x is equal to b raised to power y. Now using this result we can write x is equal to b raised to power y as y is equal to log of x to the base b. So this implies y is equal to log of x to the base b. Now we replace y by f inverse of x and we get f inverse of x is equal to log of x to the base b. We were given an exponential function f of x which is equal to b raised to power x and its inverse that is f inverse of x is given by log of x to the base b. Thus we say that logarithmic function is the inverse of the corresponding exponential function. For example if we are given any function f of x that is 2 raised to power x then as the result of definition of the formula of inverse function we can find its inverse that is f inverse of x that is given by log of x to the base 2. Thus we can convert logarithms into exponential form and exponentials into logarithmic form using relationship that is y is equal to log of x to the base b is equivalent to b raised to power y is equal to x. Now we shall discuss methods to convert exponential into logarithms and vice versa. Now we shall discuss four methods. First to convert exponential form y is equal to b raised to power x in logarithms. Second to convert exponential form y is equal to a into b raised to power x in logarithms. Third to convert logarithmic form y is equal to log of x to the base b in exponential form and fourth to convert logarithmic form in exponential form that is y is equal to a into b raised to power x. Now first of all we shall discuss how to convert exponential into logarithms. We are given an exponential function that is y is equal to b raised to power x we can convert it into logarithms by using the relationship that is y is equal to log of x to the base b is equivalent to b raised to power y is equal to x so logarithmic form of the function y is equal to b raised to power x will be given by log of y to the base b is equal to x for example if we want to express y is equal to 2 raised to power x in logarithmic form by using this result we can write it in logarithmic form as log of y to the base 2 is equal to x also if we are given any exponential function of the form y is equal to a into b raised to power x and we need to convert this function into logarithmic form we first write the given exponential expression in the form y upon a is equal to b raised to power x now we change this exponential function to logarithmic form using the same relationship so we get log of y upon a to the base b is equal to x for example express y is equal to 2 into 3 raised to power x into logarithmic form now here first of all we shall divide by 2 on both sides of the equation y is equal to 2 into 3 raised to power x and we get y by 2 is equal to 2 by 2 into 3 raised to power x which implies that y by 2 is equal to 3 raised to power x now using the same relationship we can write this equation as log of y by 2 to the base 3 is equal to x which is the required equation in logarithmic form equivalent to the given equation in exponential form which will now discuss how to convert logarithmic expression into exponential form if we are given any logarithmic function that is y is equal to log of x to the base b then we can convert it into exponential form by using this relationship that is y is equal to log of x to the base b is equivalent to b raised to power y is equal to x thus the exponential form of the logarithmic function y is equal to log of x to the base b is given by b raised to power y is equal to x for example if we need to convert y is equal to log of x to the base 3 in exponential form then using this relation we get y is equal to log of x to the base 3 is equivalent to 3 raised to power y is equal to x and this is the required function in exponential form now we are going to discuss conversion of given logarithmic function in exponential form that is y is equal to a into b raised to power x for this let us consider the following example that is express logarithmic expression log of y upon 3 to the base 7 is equal to x in exponential form y is equal to a into b raised to power x here we are given the logarithmic expression that is log of y upon 3 to the base 7 is equal to x and this expression is of the form log of y upon a to the base b is equal to x now using this relationship we can write this expression as y upon a is equal to b raised to power x which further implies that y is equal to a into b raised to power x so here we see that a is given as 3 and b is 7 and we can write log of y upon 3 to the base 7 is equal to x as y upon 3 is equal to 7 raised to power x and we multiply by 3 on both sides we get y by 3 into 3 is equal to 7 raised to power x into 3 which implies that y is equal to 7 raised to power x into 3 or we can also write it as y is equal to 3 into 7 raised to power x which is of the form y is equal to a into b raised to power x now we shall discuss solving problems using the relationship y is equal to log of x to the base b is equivalent to b raised to power y is equal to x now we shall consider the following example which says solve for x and the logarithmic expression given to us is log of 125 to the base x is equal to 3 now we will solve this expression for x using this relationship using this relationship this expression can be written as x raised to power 3 is equal to 125 and we can also write it as x raised to power 3 is equal to 5 raised to power 3 which implies that x is equal to 5 which is the required answer thus the value of x is 5 thus in this session we have discussed the inverse relationship between exponential and logarithmic functions and the use of this relationship to solve problems involving logarithms and exponents this completes our session hope you enjoyed this session