I found all those Lux builds with high scores unsatisfiting and wanted to make build for friend, show her how I play Lux. This is not how you should play Lux on ranked it's good for casual play in normal. You wouldn't pick Lux on ranked anyway. She might be good in solo line but only in early phase of the game, and will be domineted in mid by casters like Annie, Anivia etc.

Pros:
- some nice cc skills
- awesome ulti every 24 sec
- huge range on ulti
- can scout bushes with Lucent Singularity
- greate farming ability

Cons:
- runs out of mana quickly if not played smart
- squishy, cant really run away in later game
- you need some skills to play her well
- not the best pick for ranked games

Here's Lux history:
Born to the prestigious Crownguards, the paragon family of Demacian service, Luxanna was destined for greatness. She grew up as the family's only daughter, and she immediately took to the advanced education and lavish parties required of families as high profile as the Crownguards. As Lux matured, it became clear that she was extraordinarily gifted. She could play tricks that made people believe they had seen things that did not actually exist. She could also hide in plain sight. Somehow, she was able to reverse engineer arcane magical spells after seeing them cast only once. She was hailed as a prodigy, drawing the affections of the Demacian government, military, and citizens alike.

As one of the youngest women to be tested by the College of Magic, she was discovered to possess a unique command over the powers of light. The young Lux viewed this as a great gift, something for her to embrace and use in the name of good. Realizing her unique skills, the Demacian military recruited and trained her in covert operations. She quickly became renowned for her daring achievements; the most dangerous of which found her deep in the chambers of the Noxian High Command. She extracted valuable inside information about the Noxus-Ionian conflict, earning her great favor with Demacians and Ionians alike. However, reconnaissance and surveillance was not for her. A light of her people, Lux's true calling was the League of Legends, where she could follow in her brother's footsteps and unleash her gifts as an inspiration for all of Demacia.

''Her guiding light makes enemies wary, but they should worry most when the light fades.''
-- Garen, The Might of Demacia

Greater Quintessence of Ability Power, Greater Glyph of Ability Power - to punish your enemies harder in early phase of the game.
Greater Mark of Magic Penetration + Archaic Knowledge gives you enough magic penetration to kill most players. If they stack on magic resistence you should think about buying Void Staff
Greater Seal of Scaling Mana Regeneration - Lux is mana hungry so this helps you a little.

I didn't pick Greater Glyph of Ability Power because items and masteries gives you 39% cd which is enough.
I'm going with similar setting for most casters. You can use those runes for other casters if you want to try them.

9 points in offence and 21 in utility is standard for caster characters. You need to put points in offence to get some ability power, cooldown reduction on abilities and magic penetration (most usefull mastery for casters in offence tree).
You put 21 points in utility to get more mana regeneration, cooldown reduction, and experiance gaind from Awareness . Belive me or not those 5% experince gives you 6 lvl much faster than your opponent which could get you some kills.

Flash is most powerfull and overused spell in Leauge of Legends. You will see almost every body use Flash on higher levels. Flash lets u teleport in a small area. Grate for chasing and escaping. F.e. you chase somebody who runs into minion wave. Normaly it would stop you, but you can jump over them with Flash. Flash lets you teleport much further than range shows when you "jump over" parts of terrain. The rule is that the range must finish just ater the middle of this terrain obstruction.
You can see here some uses of Flash in the end of this video how to jump over terrain:
http://www.youtube.com/watch?v=CoiVmU4ns3s
* this video is not mine, I'm just useing it for references

Teleport - Grate spell that lets you teleport to any point of map if there's friendly minion or friendly tower. Many people forget that you can also teleport to friendly wards! This spell is very important for Lux. She's not mobile character. If she's on middle line you can use it to get back to fight quickly after regenerating health/mana and buying items. This is very important to outfarm your enemy on midlle line (most of them won't have teleport and needs to go back "on foot") so they wont dominate whole game.

Other good choices:
Ignite
Clairvoyance
Exhaust
Clarity

Why I dont pick Clarity, when you can find this spell in most builds?
You don't really have to spam spells all the time with Lux to farm efficiently. You need to cast one Lucent Singularity on each minion wave so you hit all 6 of them and finish them with nomral atacks. Clarity is usefull only in early phase of the game and you won't really use it later. You have teleport spell so you can allways go back to base and use teleport to return to your line quickly. Spaming spells with Lux is not that grate idea because she's got long cooldowns on abilities. If you cast all your skills you will get vournable and any smart player will use this oportunity to kill you or at least deal tons of dmg to you.

Illumination - Lux's damaging spells charge the target with energy for 6 seconds. Lux's next attack ignites the energy, dealing 20-190 (depending on Lux's level) magic damage to the target.
It's Lux passive and this scales grate with Lich Bane. If you hurt your enemy with spell don't forget to auto attack them once so u deal more dmg to your enemy.
Normaly you will go with this combo:
Light Binding -> Lucent Singularity (before you detonate it u use auto atack) -> detonate Lucent Singularity -> auto attack -> Finales Funkeln -> auto attack (if not out of range)
With Lich Bane passive this combo is devastating.

Light Binding - Lux releases a sphere of light that binds and deals damage to up to two enemy units, dealing 50% effect to the 2nd unit struck.
Fires a ball of light towards a target location, binding and damaging up to 2 enemy units. The first target takes 60/110/160/210/260 (+0.7) magic damage and is bound for 2 seconds. The second target receives 50% of the effect.
Cost: 60/75/90/105/120 Mana
Range: 10000
Like most of her spell this one has long range. Using this skill requires practice. Dont use this spell on minion wave - it's a huge waste. You dont know if you will get ganked so better save this spell for champion fights. Theres a trick with this spell. Normally your enemy will hide behind minion wave, remember that your spell atack up to two targets. So if there's posibility that only one minion stands before enemy champion - cast this spell it will bind minion and champion standing behind him.

Prismatic Barrier- Lux throws her wand and bends the light around any friendly target it touches, protecting them from enemy damage.
Lux throws her wand to the target location and back, protecting herself and all allied champions it touches from 60/85/110/135/160 (+0.35) damage for seconds.
Cost: 60/60/60/60/60 Mana
Range: 10000
Support kind of spell that gives you some kind of shield. I've never seen it beeing any usefull in team fights. Its not like Morgana's Spell shield that gives you immunity to snare effects, it just shields you from one standard attack. The only use I find this spell is when playing against Khartus. This spell saved me few times from his ultimate. I cast it only to get passive from Lich Bane.

Lucent Singularity- Fires an anomaly of twisted light to an area, which slows nearby enemies. Lux can detonate it to damage enemies in the area of effect.
Creates a zone that slows enemy units by 20/24/28/32/36% (zone lasts 5 seconds). Can be detonated to damage enemies in the area for 60/105/150/195/240 (+0.6) magic damage.
Cost: 70/85/100/115/130 Mana
Range: 1100
Awsome skill that has many uses. First to farm minion waves. Later in game you will be able to kill all waves only with this spell. You can use this spell to check bushes for enemies (its better than just going there and die in an ambush). It lets you escape or helps with chasing (because of slow effect). Don't forget to detonate this spell after you cast. I find myself in heat of the battle sometimes forgetting to detonate it. First cast sends slowin ball, secound cast detonates it. It can detonate on its own but you wont find anyone so stupid to stay in range :)

Finales Funkeln - After gathering energy, Lux fires a beam of light that deals damage to all targets in the area. In addition, triggers Lux's passive ability and refreshes the Illumination debuff duration.
After a short delay, Lux fires a stream of light in front of her, dealing 300/400/500 (+0.75) to all enemy units in the area. In addition, ignites the target for Illumination damage if under its effect, and refreshes the Illumination debuff.
Cost: 100/150/200 Mana
Range: 3000
One of the best ultimates out there. Gives you lot of fun. The cooldown is really quick and with all that cooldown reduction items you can cast it every 24 secounds on 16 lvl (SIC!). So belive me... don't be scared of using your ultimate. 24 secounds its really fast. This spell has huge range. So big you can't really see the end. I'ts good to know how far you can shoot with it so when playing vs bots, try aiming your ultimate, scroll map untill you find end of aiming target and shoot it.

You should master Lucent Singularity as soon as possible, picking Light Binding on 2nd lvl and Prismatic Barrier on 4. After mastering Lucent Singularity, put points into Light Binding and last points on Prismatic Barrier. Dont forget to put points into Finales Funkeln (ultimate) on 6lvl, 11 lvl and 16 lvl!

That item sequence is a standard one I would use in normal games. The thing is after you'll get some experience playing LoL you will adapt your items to the game. Against "heavy" Teams you will try to get tanky too with Rod of Ages and Rylas Scepter just to survive a little bit longer. If enemy starts building Magic Resistance early in game (or because you have 2-3 other players in team that focus on magic dmg) you will need to buy lot of magic penetration with Sorceress boots and Void Staff. Nevertheless I'll always buy Morello's Evil Tome because it gives me lot of cooldown reduction on abilites, mana regen and ability power.
Just if you didn't know. The max lvl of cooldown reduction is 40% You can't get any more so buing more items with CD(cooldown reduction) won't give you more CD than 40%.

Lux is not the best caster - more of caster/support with her cc(crowd control) skills and shield. So building her on full ap makes no sense or you will be focused all the time. You will be able to cast one/two skills before some carry takes you down. I think building her with more survability is better.

Starting with Doran's Ring - grate starting item you can buy right away that gives you 100 health (its a lot in early game!), mana regen and 15 ability power. Its only 5 ablity power less than Amplifying Tome, but you get that health and mana regen!
After that you should buy Boots - to be able to run from enemies skill. If game looks to hard you could consider buying another Doran's Ring.
Ionian Boots of Lucidity and Fiendish Codex should be bought as soon as possible. Those two + masteries gieves you a lot of cooldown reduction so you could farm efficiently. You can wait with upgrading to Morello's Evil Tome and try to buy Rod of Ages first. That gives you more survability and ability power.
You're now probably in later phase of middle game so you should think about getting more ap.
Rabadon's Deathcap is best choice of every castaer.
Not many games persist until late phase when you can finish your build, but if it does go with Lich Bane and Zhonya's Hourglass. Dont Forget - if you had finished buying all the items you can always buy eliksirs (you need to be alive, and you dont need space in inventory - thwy will be consumed imidiately)

This build is made for casual gamers. Haven't try it in ranked, because Lux is not a good pick for ranked game. Leave any comments - I'll appriciate that.