Alternator Not holding charge

Vehicles: any failing to keep its battery charged.

A vehicle unable to charge its own battery has one of 4 problems:
(a) alternator failure
(b) voltage regulator failure
(c) battery failure
(d) wiring problem between battery and alternator/voltage regulator.

One most modern vehicles (including 2002 Lexus RX300 - 2WD and AWD), the voltage regulator is an integral component of the alternator and is not separately serviceable.

In the US, one can get a free "charging system diagnosis" from the popular auto parts chains: AutoZone, Advance Auto Parts & Pep Boys. You needn't remove any parts from the car to get this diagnostic, since they can attach a diagnostic meter to the charging system in the parking lot. This diagnostic aid will tell you exactly which component has failed - battery, alternator or voltage regulator.

In case the vehicle is immobile, one can DIY (do it yourself) the diagnosis.
(a) inspect the wiring for corrosion/loose connections/loose connectors/etc.
(b) check alternator belt/pulley - if drive belt is properly turning the alternator pulley (no slippage/misrouting/etc.), then the mechanical tests are complete, and you'll need to continue testing the electrical performance of the charging system components.
(c) first component to test: battery
DIY test 1: remove battery from car and put battery on a 120VAC automotive battery charger and charge it fully (or just check it in the car with motor off, since the car's charging system is a type of automotive battery charger).
test parameter: a fully charged lead-acid automotive battery should read 12.45 volts on a VOM/DMM
DIY test 2: disassemble battery out of car after driving it to one of the auto parts chain stores (Advance/AutoZone/Pep Boys) for a free battery test. These testers will test the battery under load, which is not possible with just a DMM.
(d) if wiring is good, and battery tests good under load, then
the failed component is the alternator/voltage regulator - by process of elimination.
(e) DIY test 3: direct alternator/voltage regulator test (car must start and idle successfully to perform this test)
Start the car, and put a VOM/DMM across the terminals of the battery. Since the car is running, you'll be reading the output voltage of the alternator and not the output voltage of the battery. The acceptable ranges for alternator/voltage regulator output are:

ALTERNATOR CHARGING VOLTAGE

Most alternators that are charging properly should produce a voltage of about 13.8 to 14.2 volts at idle with the lights and accessories off. Always refer to the vehicle manufacturer's specifications. Many Asian vehicles, for example, have higher charging voltages of around 15 volts.

When the engine is first started, the charging voltage should rise quickly to about two volts above base battery voltage, then taper off, leveling out at the specified voltage.

The exact charging voltage will vary according to the battery's state of charge, the load on the vehicle's electrical system, and temperature. The lower the temperature the higher the charging voltage, and the higher the temperature the lower the charging voltage. The "normal" charging voltage on a typical application might be 13.9 to 15.1 volts at 77 degrees F. But at 20 degrees F. below zero, the charging voltage might be 14.9 to 15.8 volts. On a hot engine on a hot day, the normal charging voltage might drop to 13.5 to 14.3 volts.

Here are the full specs for installation of the 2002 Lexus RX300 alternator - you may be able to check these specs yourself (with a torque wrench), or pass them along to your mechanic.

Note: the VIN 8th digit should be "F" for the 2002 Lexus RX300 (2WD & AWD)

2002 Lexus RX300 (2WD and AWD) - 3.0L Engine, VIN "F" SFI DOHC

Alternator

Drive belt. Tension the belt to 170-180 lbs. for a new belt or 95-135 lbs. for a used belt.
Adjusting alternator lockbolt. Tighten the bolt to 13 ft.-lbs. (18 Nm).
Alternator pivot bolt. Tighten the bolt to 41 ft.-lbs. (56 Nm).

Glossary of acronyms
--------------------------------
DIY = do it yourself
DMM = Digital Multimeter
DOHC = Dual Overhead Cam
SFI = Sequential Fuel Injection
VOM = Volt Ohmmeter

References
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How to test a Car Alternator - todayifoundout.com

Alternator & Charging System Checks - aa1car.com