I have some rules that use both the log and run directives. In the run directive, I "manually" open and write the log file.
For instance:
rule compute_RPM:
    input:
        counts_table = source_small_RNA_counts,
        summary_table = rules.gather_read_counts_summaries.output.summary_table,
        tags_table = rules.associate_small_type.output.tags_table,
    output:
        RPM_table = OPJ(
            annot_counts_dir,
            "all_{mapped_type}_on_%s" % genome, "{small_type}_RPM.txt"),
    log:
        log = OPJ(log_dir, "compute_RPM_{mapped_type}", "{small_type}.log"),
    benchmark:
        OPJ(log_dir, "compute_RPM_{mapped_type}", "{small_type}_benchmark.txt"),
    run:
        with open(log.log, "w") as logfile:
            logfile.write(f"Reading column counts from {input.counts_table}\n")
            counts_data = pd.read_table(
                input.counts_table,
                index_col="gene")
            logfile.write(f"Reading number of non-structural mappers from {input.summary_table}\n")
            norm = pd.read_table(input.summary_table, index_col=0).loc["non_structural"]
            logfile.write(str(norm))
            logfile.write("Computing counts per million non-structural mappers\n")
            RPM = 1000000 * counts_data / norm
            add_tags_column(RPM, input.tags_table, "small_type").to_csv(output.RPM_table, sep="\t")
For third-party code that writes to stdout, maybe the redirect_stdout context manager could be helpful (found in https://stackoverflow.com/a/40417352/1878788, documented at
https://docs.python.org/3/library/contextlib.html#contextlib.redirect_stdout).
Test snakefile, test_run_log.snakefile:
from contextlib import redirect_stdout
rule all:
    input:
        "test_run_log.txt"
rule test_run_log:
    output:
        "test_run_log.txt"
    log:
        "test_run_log.log"
    run:
        with open(log[0], "w") as log_file:
            with redirect_stdout(log_file):
                print(f"Writing result to {output[0]}")
                with open(output[0], "w") as out_file:
                    out_file.write("result\n")
Running it:
$ snakemake -s test_run_log.snakefile
Results:
$ cat test_run_log.log 
Writing result to test_run_log.txt
$ cat test_run_log.txt 
result