AppleScript doesn't make it easy:
- logonly logs while running in AppleScript Editor or when running via- osascript(to- stderrin that case) - the output will be lost in other cases, such as when applications run a script with the- NSAppleScriptCocoa class.
 
- logonly accepts one argument; while it does accept any object type, it doesn't make it easy to get a meaningful representation of non-built-in types: try- log meto get information about the script itself, for instance; frequently,- log (get properties of <someObj>)must be used to get meaningful information;  note the cumbersome syntax, which is required, because just using- log properties of <someObj>typically merely prints the name of the reference form instead of the properties it points to (e.g,- log properties of meuselessly outputs just- (*properties*)).
 
- In general, AppleScript makes it very hard to get meaningful text representations of objects of non-built-in types: - <someObj> as text(same as:- <someObj> as string) annoyingly breaks - throws a runtime error - for such objects; try- me as text.
 
Below are helper subroutines that address these issues:
- dlog()is a subroutine that combines deriving meaningful text representations of any objects with the ability to write to multiple log targets (including syslog and files) based on a global config variable.
- toString()(effectively embedded in- dlog()) is a subroutine that takes a single object of any type and derives a meaningful text representation from it.
Tip of the hat to @1.61803; his answer provided pointers for implementing the various logging targets.
Examples:
  # Setup: Log to syslog and a file in the home dir.
  #        Other targets supported: "log", "alert"
  #        Set to {} to suppress logging.
set DLOG_TARGETS to { "syslog", "~/as.log" } 
  # Log properties of the front window of frontmost application.
dlog(front window of application (path to frontmost application as text))
  # Log properties of own front window; note the *list* syntax for multiple args.
dlog({"my front window: ", front window})
  # Get properties of the running script as string.
toString(me) # ->, e.g.: [script name="sandbox"] {selection:insertion point after character 2475 of text of document "sandbox2.scpt", frontmost:true, class:application, name:"AppleScript Editor", version:"2.6"}
See the source-code comments above each subroutine for details.
dlog() source code
    # Logs a text representation of the specified object or objects, which may be of any type, typically for debugging.
    # Works hard to find a meaningful text representation of each object.
    # SYNOPSIS
    #   dlog(anyObjOrListOfObjects)
    # USE EXAMPLES
    #   dlog("before")  # single object
    #     dlog({ "front window: ", front window }) # list of objects
    # SETUP
    #   At the top of your script, define global variable DLOG_TARGETS and set it to a *list* of targets (even if you only have 1 target).
    #     set DLOG_TARGETS to {} # must be a list with any combination of: "log", "syslog", "alert", <posixFilePath>
    #   An *empty* list means that logging should be *disabled*.
    #   If you specify a POSIX file path, the file will be *appended* to; variable references in the path
    #   are allowed, and as a courtesy the path may start with "~" to refer to your home dir.
    #   Caveat: while you can *remove* the variable definition to disable logging, you'll take an additional performance hit.
    # SETUP EXAMPLES
    #    For instance, to use both AppleScript's log command *and* display a GUI alert, use:
    #       set DLOG_TARGETS to { "log", "alert" }
    # Note: 
    #   - Since the subroutine is still called even when DLOG_TARGETS is an empty list, 
    #     you pay a performancy penalty for leaving dlog() calls in your code.
    #   - Unlike with the built-in log() method, you MUST use parentheses around the parameter.
    #   - To specify more than one object, pass a *list*. Note that while you could try to synthesize a single
    #     output string by concatenation yourself, you'd lose the benefit of this subroutine's ability to derive
    #     readable text representations even of objects that can't simply be converted with `as text`.
    on dlog(anyObjOrListOfObjects)
        global DLOG_TARGETS
        try
            if length of DLOG_TARGETS is 0 then return
        on error
            return
        end try
        # The following tries hard to derive a readable representation from the input object(s).
        if class of anyObjOrListOfObjects is not list then set anyObjOrListOfObjects to {anyObjOrListOfObjects}
        local lst, i, txt, errMsg, orgTids, oName, oId, prefix, logTarget, txtCombined, prefixTime, prefixDateTime
        set lst to {}
        repeat with anyObj in anyObjOrListOfObjects
            set txt to ""
            repeat with i from 1 to 2
                try
                    if i is 1 then
                        if class of anyObj is list then
                            set {orgTids, AppleScript's text item delimiters} to {AppleScript's text item delimiters, {", "}} # '
                            set txt to ("{" & anyObj as string) & "}"
                            set AppleScript's text item delimiters to orgTids # '
                        else
                            set txt to anyObj as string
                        end if
                    else
                        set txt to properties of anyObj as string
                    end if
                on error errMsg
                    # Trick for records and record-*like* objects:
                    # We exploit the fact that the error message contains the desired string representation of the record, so we extract it from there. This (still) works as of AS 2.3 (OS X 10.9).
                    try
                        set txt to do shell script "egrep -o '\\{.*\\}' <<< " & quoted form of errMsg
                    end try
                end try
                if txt is not "" then exit repeat
            end repeat
            set prefix to ""
            if class of anyObj is not in {text, integer, real, boolean, date, list, record} and anyObj is not missing value then
                set prefix to "[" & class of anyObj
                set oName to ""
                set oId to ""
                try
                    set oName to name of anyObj
                    if oName is not missing value then set prefix to prefix & " name=\"" & oName & "\""
                end try
                try
                    set oId to id of anyObj
                    if oId is not missing value then set prefix to prefix & " id=" & oId
                end try
                set prefix to prefix & "] "
                set txt to prefix & txt
            end if
            set lst to lst & txt
        end repeat
        set {orgTids, AppleScript's text item delimiters} to {AppleScript's text item delimiters, {" "}} # '
        set txtCombined to lst as string
        set prefixTime to "[" & time string of (current date) & "] "
        set prefixDateTime to "[" & short date string of (current date) & " " & text 2 thru -1 of prefixTime
        set AppleScript's text item delimiters to orgTids # '
        # Log the result to every target specified.
        repeat with logTarget in DLOG_TARGETS
            if contents of logTarget is "log" then
                log prefixTime & txtCombined
            else if contents of logTarget is "alert" then
                display alert prefixTime & txtCombined
            else if contents of logTarget is "syslog" then
                do shell script "logger -t " & quoted form of ("AS: " & (name of me)) & " " & quoted form of txtCombined
            else # assumed to be a POSIX file path to *append* to.
                set fpath to contents of logTarget
                if fpath starts with "~/" then set fpath to "$HOME/" & text 3 thru -1 of fpath
                do shell script "printf '%s\\n' " & quoted form of (prefixDateTime & txtCombined) & " >> \"" & fpath & "\""
            end if
        end repeat
    end dlog
toString() source code
    # Converts the specified object - which may be of any type - into a string representation for logging/debugging.
    # Tries hard to find a readable representation - sadly, simple conversion with `as text` mostly doesn't work with non-primitive types.
    # An attempt is made to list the properties of non-primitive types (does not always work), and the result is prefixed with the type (class) name
    # and, if present, the object's name and ID.
    # EXAMPLE
    #       toString(path to desktop)  # -> "[alias] Macintosh HD:Users:mklement:Desktop:"
    # To test this subroutine and see the various representations, use the following:
    #   repeat with elem in {42, 3.14, "two", true, (current date), {"one", "two", "three"}, {one:1, two:"deux", three:false}, missing value, me,  path to desktop, front window of application (path to frontmost application as text)}
    #       log my toString(contents of elem)
    #   end repeat
    on toString(anyObj)
        local i, txt, errMsg, orgTids, oName, oId, prefix
        set txt to ""
        repeat with i from 1 to 2
            try
                if i is 1 then
                    if class of anyObj is list then
                        set {orgTids, AppleScript's text item delimiters} to {AppleScript's text item delimiters, {", "}}
                        set txt to ("{" & anyObj as string) & "}"
                        set AppleScript's text item delimiters to orgTids # '
                    else
                        set txt to anyObj as string
                    end if
                else
                    set txt to properties of anyObj as string
                end if
            on error errMsg
                # Trick for records and record-*like* objects:
                # We exploit the fact that the error message contains the desired string representation of the record, so we extract it from there. This (still) works as of AS 2.3 (OS X 10.9).
                try
                    set txt to do shell script "egrep -o '\\{.*\\}' <<< " & quoted form of errMsg
                end try
            end try
            if txt is not "" then exit repeat
        end repeat
        set prefix to ""
        if class of anyObj is not in {text, integer, real, boolean, date, list, record} and anyObj is not missing value then
            set prefix to "[" & class of anyObj
            set oName to ""
            set oId to ""
            try
                set oName to name of anyObj
                if oName is not missing value then set prefix to prefix & " name=\"" & oName & "\""
            end try
            try
                set oId to id of anyObj
                if oId is not missing value then set prefix to prefix & " id=" & oId
            end try
            set prefix to prefix & "] "
        end if
        return prefix & txt
    end toString