EXPONENTS AND LOGARITHMS

Transcription

1 Mthemtis Stdrd Level for IB Diplom Eerpt EXPONENTS AND LOGARITHMS WHAT YOU NEED TO KNOW The rules of epoets: m = m+ m = m ( m ) = m m m = = () = The reltioship etwee epoets d rithms: = g where is lled the se of the rithm The rules of rithms: = = 0 Epoets d rithms

2 Mthemtis Stdrd Level for IB Diplom Eerpt The hge of se rule: = There re two ommo revitios for rithms to prtiulr ses: 0 is ofte writte s e is ofte writte s l The grphs of epoetil d rithmi futios: y y = Ae y y y = y = C + Ae C + A A y = C epoetil growth epoetil dey rithm EXAM TIPS AND COMMON ERRORS You must kow wht you ot do with rithms: ( + y) ot e simplified; it is ot + y (e + e y ) ot e simplified; it is ot + y ( ) is ot, wheres = l g e l og e e e ot e + Epoets d rithms

3 Mthemtis Stdrd Level for IB Diplom Eerpt. SOLVING EXPONENTIAL EQUATIONS WORKED EXAMPLE. Solve the equtio + =, givig your swer i the form. ( ) ( ( ) + l g )= ( ( )= ( ) Sie the ukow is i the power, we tke rithms of eh side. We the use the rules of rithms to simplify the epressio. First use () = + A ommo mistke is to sy tht ( + ) = ( + ). l g +( + + ) l og= ( l g = l g l og= + ( l g l og)= + = 0 = We ow use k = k to get rid of the powers. Epd the rkets d ollet the terms otiig o oe side. Use the rules of rithms to write the solutio i the orret form: = = Prtie questios.. Solve the equtio, givig your swer i the form re itegers. where d. Solve the equtio, givig your swer i the form p rtiol umers. q where p d q re. Solve the equtio rtiol umers. p =, givig your swer i the form q where p d q re Epoets d rithms

4 Mthemtis Stdrd Level for IB Diplom Eerpt. SOLVING DISGUISED QUADRATIC EQUATIONS WORKED EXAMPLE. Fid the et solutio of the equtio + =. = ( ) We eed to fi d sustitutio to tur this ito qudrti equtio. First, epress + i terms of : ( ) Look out for term, whih e rewritte s ( ). Let y =. The y y = 0 ( yy )( y ) y = o y = After sustitutig y for, this eomes stdrd qudrti equtio, whih e ftorised d solved. Disguised qudrti equtios my lso e eoutered whe solvig trigoometri equtios, whih is overed i Chpter. = or = = is impossile sie > 0 for ll. l g = l g = = Rememer tht y =. With disguised qudrti equtios, ofte oe of the solutios is impossile. Sie is i the power, we tke rithms of oth sides. We the use k = k to get rid of the power. Prtie questios.. Solve the equtio + = 0.. Fid the et solutio of the equtio e 6e =. 6. Solve the simulteous equtios e + y = 6 d e + e y =. Epoets d rithms

5 Mthemtis Stdrd Level for IB Diplom Eerpt. LAWS OF LOGARITHMS WORKED EXAMPLE. If =, y = d z =, write + y 0.z + s sigle rithm. 0. og 0 =. + We eed to rewrite the epressio s sigle rithm. I order to pply the rules for omiig rithms, eh must hve o oeffi iet i frot of it. So we fi rst eed to use k = k. 0. = + = + We ow use + y = ( y) d = y y = l + og = We lso eed to write s rithm so tht it the e omied with the fi rst term. Sie 0 = 00, we write s 00. Rememer tht o its ow is tke to me 0. Prtie questios. 7. Give =, y = d z =, write y + z s sigle rithm. 8. Give =, = y d = z, fid epressio i terms of, d for 0 y z. 9. Give tht + =, epress i terms of. 0. Give tht l y = + l, epress y i terms of.. Cosider the simulteous equtios e + e y = 800 l + l y = () For eh equtio, epress y i terms of. () Hee solve the simulteous equtios. Epoets d rithms

6 Mthemtis Stdrd Level for IB Diplom Eerpt. SOLVING EQUATIONS INVOLVING LOGARITHMS WORKED EXAMPLE. Solve the equtio l g 9. = = Therefore g 9 g = 9 ( g ) = 9 ( ) = 9 We wt to hve rithms ivolvig just oe se so tht we pply the rules of rithms. Here we use the hge of se rule to tur s with se ito s with se. (Altertively, we ould hve tured them ll ito se isted.) Multiply through y to get the terms together. Mke sure you use rkets to idite tht the whole of is eig squred, ot just ; ( ) is ot equl to, ut would e. = Tke the squre root of oth sides; do t forget the egtive squre root. So or = Use = to udo the s. = 8 = 8 Prtie questios.. Solve the equtio ( 6).. Solve the equtio l g ( ) =, givig your swers i simplified surd form. Mke sure you hek your swers y sustitutig them ito the origil equtio.. Solve the equtio.. Solve the equtio ( ) 6 ( ). 6 Epoets d rithms

7 Mthemtis Stdrd Level for IB Diplom Eerpt. PROBLEMS INVOLVING EXPONENTIAL FUNCTIONS WORKED EXAMPLE. Whe up of te is mde, its temperture is 8 C. After miutes the te hs ooled to 60 C. Give tht the temperture T ( C) of the up of te deys epoetilly ordig to the futio T = A + Ce 0.t, where t is the time mesured i miutes, fid: () the vlues of A d C (orret to three sigifit figures) () the time it tkes for the te to ool to 0 C. () Whe t 0 A + C Whe t 6 A Ce () () gives C e So C = =. ( SF) 06 e = ( ) ( ) ( ) The, from (): A = 8 C = 8. = 9.6 SF ( ) Sustitute the give vlues for T (temperture) d t (time) ito T = A + Ce 0.t, rememerig tht e 0 =. Note tht A is the log-term limit of the temperture, whih e iterpreted s the temperture of the room. () Whe T = 0: t e = t = ( e 0t l )= l t = l. t = 8. 6 miutes Now we sustitute for T, A d C. Sie the ukow t is i the power, we tke rithms of oth sides d the el e d l usig ( ) =. Rememer tht l mes e. Prtie questios. 6. The mout of rett, V (grms), i hemil retio deys epoetilly ordig to the futio V = M + Ce 0.t, where t is the time i seods sie the strt of the retio. Iitilly there ws. g of rett, d this hd deyed to.6 g fter 7 seods. () Fid the vlue of C. () Fid the vlue tht the mout of rett pprohes i the log term. 7. A popultio of teri grows ordig to the model P = Ae kt, where P is the size of the popultio fter t miutes. Give tht fter miutes there re 00 teri d fter miutes there re 00 teri, fid the size of the popultio fter 0 miutes. Epoets d rithms 7

8 Mthemtis Stdrd Level for IB Diplom Eerpt Mied prtie. Solve the equtio = 0.. Fid the et solutio of the equtio.. Solve the simulteous equtios l + l y = l + l y = 0. Give tht y = l l( + ) + l( ), epress i terms of y.. The grph with equtio y = l( ) psses through the poit (, l 6). Fid the vlue of. 6. () A eoomi model predits tht the demd, D, for ew produt will grow ordig to the equtio D = A Ce 0.t, where t is the umer of dys sie the produt luh. After 0 dys the demd is 000 d it is iresig t rte of per dy. (i) Fid the vlue of C. (ii) Fid the iitil demd for the produt. (iii) Fid the log-term demd predited y this model. () A ltertive model is proposed, i whih the demd grows ordig to the formul t + D B 0 l. The iitil demd is the sme s tht for the first model. (i) Fid the vlue of B. (ii) Wht is the log-term preditio of this model? () After how my dys will the demd predited y the seod model eome lrger th the demd predited y the first model? Goig for the top. Fid the et solutio of the equtio = 6, givig your swer i the form l p where p d q re itegers. lq. Give tht +, epress i terms of.. I physis eperimet, My mesured how the fore, F, eerted y sprig depeds o its etesio,. She the plotted the vlues of = l F d = l o grph, with o the horizotl is d o the vertil is. The grph ws stright lie, pssig through the poits (,.) d (, 7.). Fid epressio for F i terms of. 8 Epoets d rithms

9 Mthemtis Stdrd Level for IB Diplom Eerpt POLYNOMIALS WHAT YOU NEED TO KNOW The qudrti equtio + + = 0 hs solutios give y the qudrti formul: = The umer of rel solutios to qudrti equtio is determied y the disrimit, Δ =. If Δ > 0, there re two distit solutios. If Δ = 0, there is oe (repeted) solutio. If Δ < 0, there re o rel solutios. The grph of y = + + hs y-iterept t (0, ) d lie of symmetry t The grph of y ( p) ( q ) hs -iterepts t (p, 0) d (q, 0). The grph of y ( h) + k hs turig poit t (h, k). =. A epressio of the form ( + ) e epded quikly usig the iomil theorem: ( r r ) = The iomil oeffiiets e foud usig lultor, Psl s trigle or the formul r =! r!( r)! EXAM TIPS AND COMMON ERRORS Mke sure tht you rerrge qudrti equtios so tht oe side is zero efore usig the qudrti formul. Questios ivolvig the disrimit re ofte disguised. You my hve to iterpret them to relise tht you eed to fid the umer of solutios rther th the tul solutios. Look out for qudrti epressios i disguise. A sustitutio is ofte good wy of mkig the epressio epliitly qudrti. Polyomils 9

10 Mthemtis Stdrd Level for IB Diplom Eerpt. USING THE QUADRATIC FORMULA WORKED EXAMPLE. Solve the equtio, givig your swer i the form. = 0 Here =, = d = = ( ) ) )± ( ± 8 = ± 7 = ± 7 = = ± 7 ( ) Rerrge the equtio to mke oe side zero; the use the qudrti formul. Use the ft tht = to simplify the swer. Prtie questios.. Solve the equtio = +, givig your swer i the form.. Fid the et solutios of the equtio + =. A et solutio i this otet mes writig your swer s surd. Eve givig ll the deiml ples show o your lultor is ot et.. Solve the equtio + 8k = 6k, givig your swer i terms of k.. Usig the sustitutio u =, solve the equtio + = 0.. A field is 6 m wider th it is log. The re of the field is 0 m. Fid the et dimesios of the field. 0 Polyomils