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Inverse Function Log

Introduction:

In mathematics, if ƒ is a function from a set A to a set B, then an inverse function for ƒ is a function from B to A, Thus, if an input x into the function ƒ produces an output y, then input is y and inverse output is x. The inverse of a logarithmic function is an exponential function. If f is a function, then inverse function of f  is f-1.

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Logarithmic properties:


Multiplication property: 

          loga(AB) = logaA + logaB

Division property:

          loga`(A/B)` = logaA - logaB

Product-power rule:

          loga(Xr) = r logaX

Some other properties:

          logaa = 1

          loga1 = 0

          loga`(1/x)` =  - logax

         logax = `(log x)/(log a)` = `(ln x) / (ln a)`


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Inverse logarithm problems:


Inverse logarithmic problem:

Determine the inverse function for the given expression ln(x - 5) + 3

Solution:

we know, the function f(f-1(x)) = x  and the base is e.

f(f-1(x)) = x -------->   f(f-1(x)) = ln(f-1(x) - 5) + 3 = x

Subtracting by 3 on left and right side of the above equation.

ln(f-1(x) - 5) + 3 - 3 = x - 3

So, we get

ln(f-1(x) - 5)  = x - 3

Now take exponential form on both side then we get,                       

`e^(x - 3) ` = f-1(x) - 5

Now add by 5 on both left and right side of the equation.

So we get,    ` e^(x - 3)` + 5 = f-1(x) - 5 + 5

                        `e^(x - 3)` + 5  = f-1(x)     

                        f-1(x) =  `e^(x - 3)` + 5 

The function f(x) is higher than 5 and the inverse function of f(x) is also higher than 5. inverse function of f(x) is f-1(x). if the line y = x is symmetry means, If the point (6, 3), graph the given function,  f-1(3) = 6. That is point (3, 6) is graph of the function f-1(x) ,

                                  `e^(x - 3)` + 5  = `e^(3 - 3)` + 5

                                                  = `e^(0)` + 5                     we know e0 = 1

                                                  = 1 + 5

                                                  = 6

      Answer: Inverse function of f(x) = 6.