There's no automatic way to redirect the log that I'm aware of.
There are a few ways to get what you want, however.
First off, if you are able to use Jupytr, SAS has plugins to enable that to work with SAS, and then you can simply write in the notebook and run the code, and the results appear with your code just as you want. See Chris Hemedinger's blog post on the subject for more details.
Second, SAS Studio will support a notebook-style interface probably with the next major revision (I believe version 5.0) which will release late next year. So similarly, you would put your code and get your output in the same windows.
Finally, the third option is to do as Reeza suggested - write to a log file, then print that to the output. It's messy but possible.
Here's an example of the latter. I don't make any effort to clean it up, note, you'd probably want to remove the logging related to PROC PRINTTO and the otehr notes (or turn on NONOTE).
ods pdf file="c:\temp\test.pdf";
filename logfile temp;
proc printto log=logfile;
run;
proc sql;
select * from sashelp.class;
quit;
proc printto;
run;
data _null_;
infile logfile;
input @1 @;
call execute(cats('ods text="',trim(_infile_),'";'));
run;
ods _all_ close;