Basically, logarithmic transformations ask, a number, to what power equals another number?


 Thomasine Cole
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1 Wht i logrithm? To nwer thi, firt try to nwer the following: wht i x in thi eqution? 9 = 3 x wht i x in thi eqution? 8 = 2 x Biclly, logrithmic trnformtion k, number, to wht power equl nother number? In prticulr, log do tht for pecific number under the exponent. Thi number i clled the be. In your cle you will relly only encounter log for two be, 10 nd e. Log be 10 We write log be ten log 10 or jut log for hort nd we define it like thi: If y = 10 x then log (y) = x So, wht i log (10 x )? log (10 x ) = x How bout 10 log(x)? 10 log(x) = x More exmple: log 100 = 2 log (10 5 )= 5 The point trt to emerge tht log re relly horthnd for exponent. Log were invented to turn multipliction problem into ddition problem. Let ee why. log (10 2 ) + log (10 3 ) = 5, or log (10 5 ) Copengle, Acdemic Support Pge 1/6
2 So, clerly there prllel between the rule of exponent nd the rule of log: Tble 4. Function of log be 10. Exponent Log be 10 Exmple r! r + = log(ab) = log(a) + log(b) log(10 5 ) = log (10 2 ) + log (10 3 ) 1! = & 1 # log $! =  log(b) log & 1 # $! = log (105 )=  5 % B " % 10 " log(10 5 ) r r & A # log(10 2 ) = log (10 5 )  log = log $! = log(a) log(b) % B " (10 3 ) = 5 3 = 2 r r ( ) = log (A x ) = xlog(a) log(10 3 ) = 3log(10) = 3 (1) = 3 0 = 1 log(1) = 0 log(10) = 1 Copengle, Acdemic Support Pge 2/6
3 Nturl log, or log be e. Why e? e = / 1! + 1 / 2! + 1 / 3! +... (remember: 3! = (3)(2)(1)) e = the limit of (1 + 1 / n ) n n e = In 1864 Benjmin Peirce would write i "i = e # nd y to hi tudent: We hve not the lightet ide wht thi eqution men, but we my be ure tht it men omething very importnt.! u de u du e h the implet derivtive: = e dx dx The derivtive of e with vrible exponent i equl to e with tht exponent time the derivtive of tht exponent. We cre becue nture doe not uully go by log, but inted by nturl log. We trt our dicuion of nturl log with imilr bic definition: We write log be e ln nd we cn define it like thi: If y = e x then ln (y) = x And o, ln(e x ) = x e ln(x) = x Now we hve new et of rule to dd to the other: Tble 4. Function of log be 10 nd be e. Exponent Log be 10 Nturl Log r! r + = log(ab) = log(a) + log(b) ln(ab) = ln(a) + ln(b) 1! = & 1 # & 1 # log $! =  log(b) ln $!" =  ln(b) % B " % B r r & A # & A # = log $! = log(a) log(b) Ln $!" = ln(a) ln(b) % B " % B r r ( ) = log (A x ) = xlog(a) ln (A x ) = xln(a) 0 = 1 log(1) = 0 ln(1) = 0 log(10) = 1 ln(e) = 1 Copengle, Acdemic Support Pge 3/6
4 Exmple: ln(e 45 ) = 45 log(10 23 x ) = ln (e 46 ) = 46 x Solve the following for x: log (256/x) = 1.5 (256/x) = x = 256/ x = 8.10 Solve K = be /rt for. To get out of the exponent, tke the ln of both ide: ln(k) = ln(b /rt) ln(k/b) = /rt (rt)ln(k/b) = or = (rt)ln(b/k) & # Solve ln$ Io! = kt for I f % I f " To get I f out of the ln, put both ide n exponent of e: & # $ Io! =e kt % I f " & I f # $! =e kt % Io " I f = I o e kt Logrithm Often when exmining our dt we find tht our plot fll long n exponentil fit, which i much more complicted thn liner function. The ue of logrithm i often pplied in thi ce to linerize exponentil function. Copengle, Acdemic Support Pge 4/6
5 Grphing with logrithm Another powerful ue of logrithm come in grphing. For exmple, exponentil function re tricky to compre viully. It hrd to ee wht hppen t mll vlue nd t lrge vlue t the me time becue the function incree (or decree) o quickly. To help with thi, we ometime plot the log of function. For exmple, look t the two function in thi grph: Figure 2. A very unhelpful plot of the frequency of ome event over time. For our purpoe it doen t much mtter wht the two function re, but we cn ee tht if we grph both A nd B on the me plot, we ee tht we hve lmot no ide wht hppening below ~15 dy on the xxi nd we lmot cn t ee the plot of A becue the cle of B i o much greter. Now, tke the me two function, but thi time plot the log (be 10 in thi ce) of ech function: Figure 3. The me dt from Figure 2, preented log plot. Alredy it i eier to compre the two nd we gin more inight to the propertie of the function t both high nd low rnge. Notice lo tht the function h become liner. Copengle, Acdemic Support Pge 5/6
6 The ue of logrithm in grphing cn lo how u importnt detil in exponentil function tht my remin hidden otherwie. For exmple, look t the following dt: Figure 4. Another unhelpful plot of n exponentil function. While thi plot i not o informtive, ee wht pper if we plot the logrithm (gin, be 10 in thi ce): Figure 5. The me dt from Figure 4 preented log plot. Now we cn ee tht there re TWO ditinct procee occurring here nd tht there i unique event t dy 21. Copengle, Acdemic Support Pge 6/6
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