Does old/low charge battery affects internal clock? Totally newbie in eletronics here, but could not find the answer for this one.

When a battery is nearly-exhausted, it's voltage and or amperage dims, right?

My question is if this kind of scenario (running low battery) would affect the arduino internal clock using only it.

I'm using arduino to a device that will operate remotely (with a 9V battery), and also logs intervals using the time function.

If battery is near-exhausted, then will this skew the numbers I get?
Let's assume by "internal clock" you mean the internal oscillator, as the arduino's processor does not have a built-in real-time clock (something that is meant to keep track of real time like seconds/minutes/hours.)

The arduino is not like a motor where it runs slower if a lower voltage is applied. If the voltage goes too low, the arduino will just reset. Once it resets, the program running in it just starts over again, unless the battery voltage is so low it can't even start up.

If you're feeding it with a 9V battery, you're going through the 5V linear regulator on-board, which is not very energy efficient. Keep that in mind if your goal is to create a project that is meant to run a long time on the 9V battery. See the notes at http://www.gammon.com.au/power for tips on reducing the power used by an arduino and/or ATMega-based project.