# 1 HALF-LIFE EQUATIONS

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1 R.L. Hanna Page HALF-LIFE EQUATIONS The basic equaion ; he saring poin ; : wrien for ime: x / where fracion of original maerial and / number of half-lives, and / log / o calculae he age (# ears): age (half-life) * / age (half-life) * log Tha s reall all here is o i. The equaions reall are ha simple! The following pages have examples and explanaions of how his simple form of he equaion is he same as he equaions in he Hisorical Geolog Lab Manual ; and he same as equaion found in exbooks.

2 R.L. Hanna Page Examples. How much is lef afer half-life? The foundaion ; he basics ; where i all begins.. / where fracion of original maerial and / number of half-lives How much is lef afer half-lives? 4 How much is lef afer 3 half-lives? 3 8 How much is lef afer 4 half-lives? 4 6 How much is lef afer 5 half-lives? 5 3 /

3 R.L. Hanna Page 3 BASIC LOGARITHMS Example ; sar here: * 4 3 ** 8 4 *** 6 Logarihm noaion reads as follows: In logarihm noaion log() 4 log(4) 3 8 log(8) log(6) 4 Take he las one. log(6) 4.This means: mus be raised o he 4 h power for a final produc of 6. Tr anoher one. log5(5)?.this means: wha power of 5 ields a final produc of 5?.well ; 5*5 5.and 5*55 ; ; herefore.5*5*55 ; or ; herefore. log5(5) 3 From page.. / log / These are he basic half-life equaions! Is reall ha simple!

4 R.L. Hanna Page 4 Mos references presen equaions ha have ln (naural logarihms) and do NOT have log base.. For example, here s one of hese ln equaions derived from one of he equaions in he Hisorical Geolog Lab Manual. age halflife *ln This equaion is acuall he same equaion as he one highlighed on he previous page.see below. To conver from log base o log base e (naural log): Sar wih: / (from he firs page) ln Take he naural log of boh sides ( ) and ln() o calculae he ages, mulipl b he half-life / ln *ln ln ( ) / / / ln ln ( ) ( ) ( ) ln age (half-life)* /. age (half-life)* halflife ( ) ln age *ln( ) Therefore.. age (half-life) * halflife log age ( *ln )

5 R.L. Hanna Page 5 Anoher version of he equaion: n *ln age ( ) K n o.which is he same as he equaions highlighed on he previous page ; see below ; and K halflife n n o fracion of original maerial no amoun of paren maerial lef n oal amoun of maerial lef paren+daugher Saring wih age halflife *ln from he previous page age K *ln n ( ) o n age n *ln K no ;.and since log x -log(x) ; ; age - n *ln K n o (Rober s equaion) n *ln age ( ) K n o age (half-life) * log he basic equaion from he firs page In conclusion, he basic equaion for age (from he firs page) ; is he same as he age equaion in he Hisorical Geolog Lab Manual and is he same as he equaion found in mos exbooks See he calculaions on he nex page for examples..

6 R.L. Hanna Page 6 ln version of he Eqn Basic Eqn from page 4.50E+09 halflife K(ln)/(half-life) (half-life)*log ( / ) (-)*(/K)*ln(n o/n ) (half-life)*log (n /n o) Island n o/n () Calculaed Age of Island Calculaed Age of Island Hawaii ,39 ears 59,39 ears Maui ,03,756 ears,03,756 ears Molokai ,63,35 ears,63,35 ears Oahu ,53,44 ears,53,44 ears Kauai ,740,984 ears 4,740,984 ears Niihau ,040,488 ears 6,040,488 ears / log( /) where is he number of half-lives.(no ears) log(x) (-)*log( /x) no amoun of paren maerial lef n oal amoun of maerial (paren+daugher) ln() when rounded.he calculaions above use ln() before rounding

7 R.L. Hanna Page 7 More Basic Logarihms.. TO TEST: logb(m n ) n logb(m) log3(8) 4 log3(8) log3(9) log3(9) * 4..ha works. Firs prove his: logb(mn) logb(m) + logb(n) logb(m) Z and logb(n) W ;..where Z and W are no known b Z m and b W n and b Q mn logb(mn) logb(b Z b W ) logb(b Z+W ) Z + W logb(m) + logb(n) Then his almos prove iself! logb(m n ) n logb(m) logb(m n ) logb(m*m*m*m ;. n imes..) logb(m) + logb(m) + logb(m) +.. n imes logb(m n ) n logb(m)

8 R.L. Hanna Page 8 The followin pages are from hp://phsics.msu.edu/~wmr/log_.hm b Dr. William Roberson Assisan Professor of Phsics a Middle Tennessee Sae Universi The Rules of Logarihms Now ha ou undersand wha a logarihm means I wan o show ou a few simple mahemaical manipulaions ha can be done using logarihms. These are ofen called he Rules of Logarihms, however he should no be mserious o ou given wha we have covered in he previous secions. The firs rule is picall expressed as log(xy) log X + log Y In words his rule saes ha if I ake wo numbers X and Y and mulipl hem ogeher and hen ake he logarihm I obain he same resul as if I had added he logarihms of he wo numbers separael. Consider he simple example of 0 imes 0 3. Muliplied ogeher hese come o 0 5 and he log of 0 5 is 5 (ha is wha he lef hand side of he equaion ells us o do). Now he log of 0 is and he righ hand side of he equaion insrucs us o add his o he log of 0 3 which is 3 o ge 5. The rule works! Of course his relaion is nohing more han puing ogeher he wo ideas () ha he log of a number is he exponen of 0 required o equal ha number and () when we mulipl powers of 0 we add he exponens. The second rule of logarihms is a sraighforward exension of he firs--i saes ha log (X n ) n log X ha is, if he number X is raised o he power n he resul is he same as n imes log of X. To undersand his rule imagine ha n. In ha case log (X ) log (XX) log (X) + log (X) log X where I merel wroe ou X as XX in he firs sep and hen used he firs rule above. The exension o higher n should be obvious--r wriing i ou for n3. Finall here is a rule for division as well as for muliplicaion. This rule saes ha log(x/y) log X ; log Y

9 R.L. Hanna Page 9 Exponens and Logarihms We have seen ha when wo numbers are muliplied we jus add he corresponding exponens. You should also be geing a vague sense of how his relaion is conneced o he logarihmic naure of hearing described in he inroducor module in ha a muliplicaive relaion becomes an addiive one. Now we wan o inroduce he logarihm. The concep of a logarihm is o merel replace a number b he exponen o which 0 would have o be raised o ge ha number. For example, consider he number 00. To wha power would 0 have o be raised o ge 00? I hope ou cried ou, since Thus he logarihm of 00 is. In mahemaical erms we would wrie his as log(00). Quick now! Wha is he logarihm of 000? 0? 0.? If ou answered 3,, - award ourself a major prize. Now we come o he big leap in undersanding. Wha is he logarihm of a number like 57? This number canno be expressed in a nice eas forma of 0 o some ineger exponen. Neverheless here is sill some number ha when used as he exponen will give 57. In he old das o find his number required he use of ables of logarihms. Now das we use our rus calculaors. Ge our calculaor and r hese numbers o see for ourself. No onl will his reinforce our undersanding--i will also make sure ou know where all he appropriae buons are on our calculaor! Tpe in 57 and hen find he "log" buon on our calculaor and press i. You should be reurned he value ; This number is he logarihm of 57. If ou hink abou i his value makes sense because we know ha 57 lies beween 0 and 00 and he logs of hese wo numbers are and respecivel. Finall, if we have he log of a number how do we recover he number iself? Les coninue wih he example of 57. We now know ha is he log of 57. To ge from he log o he original number we mus use he log value as he exponen of 0. Some calculaors have a 0 x buon. In his case ener and hi he 0 x buon. Voila! You should ge 57 (or somehing pre close depending on how man significan digis ou enered). Unforunael no all calculaors have he 0 x buon. Some require ha ou ener and hen hi INV and hen he LOG buons. Make sure ou figure ou how o use our calculaor o ake he log of a number and o ge from he log value back o he number. You can ask me I've deal wih ever manner of calculaor over he semesers and I love a challenge. Tr he following self es o verif our calculaor savv.

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