To append a string to a file you will need to open the file using fopen() with the 'a' parameter. For example if your log file name is mylog.txt you would write the code like this :
// open the log file and check if the it's opened successfully
if (!($fp = fopen('mylog.txt', 'a'))) {
die('Cannot open log file');
}
// ... your code do something here
// append to log file
// make sure you add a newline character at the end of the log string
fwrite($fp, "Initiate world domination\n");
// ... do some stuff here
// ... and your code do something else here
// append another line to the log file
fwrite($fp, "World domination completed. Bwa ha ha ha...!!");