We denotethe naturallogarithmof by ratherthan So keepin mind,that is the power
to which mustbe raisedto produce That is, the followingtwo expressionsare equivalent:
The tablebelowshowsthis operation.
NaturalLogarithm
EquivalentExponentialFormA ComparisonbetweenLogarithmicFunctionsand ExponentialFunctions
Lookingat the two graphsof exponentialfunctionsabove,we noticethat both passthe horizontalline test.
This meansthat an exponentialfunctionis a one-to-onefunctionand thus has an inverse.To find a formula
for this inverse,we start with the exponentialfunction
Interchanging and
Projectingthe logarithmto the base on both sides,
Thus is the inverseof
This impliesthat the graphsof and are reflectionsof one anotheraboutthe line The figure
belowshowsthis relationship.