THE **figure 1 **below suggests that if *B* > 0 and *b 1 then the graph of y = satisfies the horizontal line test, and this implies that the function f (x) = It has an inverse.*

*To find a formula for this inverse (with x as an independent variable) we can solve the equation x = for y with a function of x. This can be done by taking the logarithm on the basis of B on both sides of this equation. This gives way to *

* = ()*

*But if we think () as an exponent to which B must be raised to produce then it is evident that (). So it can be rewritten as*

*y* =

*where we conclude that the inverse of f (x) = é (x) = x. This implies that the graphic in x = and the of y = are reflections of each other in relation to the straight line y = x. *

*We will call in logarithmic function in the base B.*

*In particular, if we take f (x) = and (x) = , and if we keep in mind that the domain of is the same as the image of f, then we get *

log_{B}(B^{x}) = x for all actual values of x b^{log x}= x to x> 0 |

In other words, the equation tells us that log functions_{B}(B^{x}) and b^{log x} cancel the effect of another when composed in any order; for example