Logarithmic Functions

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

 logB(Bx) = x for all actual values ​​of x blog x= x to x> 0

In other words, the equation tells us that log functionsB(Bx) and blog x cancel the effect of another when composed in any order; for example

Next: Explicitly and Implicitly Defined Functions