what is all the different log's

Logs don't have to have just integer bases Melody.  ln is log to base e, for example, and e is definitely not an integer!

A couple of other examples:

$${{log}}_{{\mathtt{4.5}}}{\left({\mathtt{6}}\right)} = {\mathtt{1.191\: \!268\: \!130\: \!927\: \!555\: \!2}}$$

$${{log}}_{{\mathtt{2.345}}}{\left({\mathtt{7}}\right)} = {\mathtt{2.283\: \!167\: \!287\: \!297\: \!550\: \!4}}$$

$${{log}}_{{\mathtt{3.45}}}{\left({\mathtt{1.2}}\right)} = {\mathtt{0.147\: \!226\: \!542\: \!852\: \!363\: \!6}}$$

.

There are two log keys on your calculator:

'log'  is used for logs with a base of ten; called "common logs"; created to make calculations much easier before there were hand-held calculator.

'ln'  is used for logs with a base of e (an irrational number, approximately 2.7182818); called "natural logs"; because they occur naturally when using the base ten system (used exclusively in calculus and problems requiring calculus). 

Logs can be any base  (integers I think)

eg

$$y=log_381$$

this means

$$81=3^y$$

3*3*3*3=81

so

y=4

The calculator only works in base 10 and base e but it is easy to convoret other bases to either of these.

Thanks Alan :)