MA 000 Lesson 1 Notes ( 5) How would person solve n eqution with vrible in n eponent, such s 9? (We cnnot re-write this eqution esil with the sme bse.) A nottion ws developed so tht equtions such s this one could be solved. This new function is clled logrithm or logrithmic function. Definition of logrithm or logrithmic function: Let 0, 1, nd 0 ( n positive number other thn 1 nd n positive number), then the logrithm nd the logrithmic function is defined s f ( ) log nd is equivlent to. **It is importnt to understnd tht logrithm is n eponent!** E 1: Write ech eponentil epression s logrithmic epression. (Convert from eponentil form to logrithmic form.) 5 8 ) 4 16 b) m q c) 3 7 n d e e f n 5 ( ) 1 ) 10 ) 8 ) 5000 E : Write ech logrithmic epression s n eponentil epression. (Convert from logrithmic form to eponentil form.) ) log 15 3 b) log 100 3 c) log n 3 5 r d) log 9 m e) log 5 ( q 1) f ) log n 4 1 4 b 0. E 3: Find ech logrithm. (Remember, logrithm is n eponent! You re sked to find n eponent, if it eists.) 1 ) log16 b) log 5 c) log1010000 5 d) log 81 e) log 64 f ) log 8 3 4 1/ 1
Below is grph of nd its inverse, or log. (0,1) (1,0) (, 1) If ou imgine the line =, ou cn see the smmetr bout tht line. A function nd its inverse will hve smmetr bout the line =. Below re both grphs on the sme coordinte sstem long with the line =. There is nother grph of f ( ) 1 nd the inverse, f ( ) log on pge 91 (clculus prt of tetbook). Agin, ou cn see the smmetr bout the line =. PROPERTIES OF LOGARITHMS: Let nd be n positive rel numbers nd r be n rel number. Let be positive rel number other thn 1 ( 1). Then the following properties eist.. log log log b. log log log r c. log r log d. log 1 e. log 1 0 r f. log r
The properties of logrithms re es to prove, if ou remember tht logrithm is n eponent. Logrithms behve like eponents. When multipling, eponents re dded (propert ). When dividing, eponents re subtrcted (propert b). When power is rised to nother power, the eponents re multiplied (propert c). An rel number to the first power is itself (propert d). An rel number to the zero power is 1 (propert e). To prove propert f, just put the logrithmic epression in eponentil form. E 4: Use the properties of logrithms to write the epression s sum, difference, or product of simpler logrithms. ) log (5 ) b) log b z 1 c) log5 3 65 E 5: Suppose log 3 m, log 4 n, nd log 5 r. Use the properties of logrithms to find the following. ) log 1 5 b) log 3 c) log 48 3
Your scientific 1-line clcultor will find logrithms using bse 10 (common logrithms) or bse e (nturl logrithms). If logrithm is written log (with no bse indicted), it is ssumed to be common logrithm, or bse 10 logrithm. If logrithm is written ln, it is ssumed to be nturl logrithm (bse e logrithm). Use our clcultor to pproimte ech of the following to 4 deciml plces. (Enter the number, then either the log ke or the ln ke.) ) ln b) log 49 c) ln 0.05 d) log 3. Chnge of bse Theorem for Logrithms: If is n positive number nd if nd b re positive rel numbers, 1, b 1, then logb log logb Most often, bse b is chosen to be 10 or e, if clcultor will be used to pproimte. log ln log or log log ln E 6: Use nturl logrithms to evlute the logrithm. Give n ect nswer nd n pproimtion to the nerest thousndth. ) log 0 b) log 0.5 4 1 3 c) log (0.03) d) log 5 19 1. 4
Solving some simple logrithmic equtions: E 7: Solve ech eqution. Hint: Convert to eponentil form. Remember ou cn onl find logrithms of positive numbers, so check our nswers. ) log 64 6 b) log 7 3 1 c) log 8 d) log 3( 5) 64 e) log( 5) log( ) 1 Solving some simple eponentil equtions: E 8: Solve ech eqution b using nturl logrithms (tke the nturl log of both sides). Approimte to four deciml points, if needed. k ) 6 15 b) e 4 c) 4 6 3 1 5
Applied problems: E 9: Leigh plns to invest $1000 into n ccount. Find the interest rte tht is needed for the mone to grow to $1500 in 8 ers if the interest is compounded continuousl. E 10: The mgnitude of n erthquke, mesured on the Richter scle, is given b RI ( ) log I where I is the mplitude registered on seismogrph locted 100 km from the I0 epicenter of the erthquke nd I 0 is the mplitude of certin smll size erthquke. Find the Richter scle rting of erthquke with the following mplitude. (Round to the nerest tenth.) 5000I 0 6