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&iOFXOR�, Autoria: Jeanne Dobgenski Tema 04 Logaritmos 7HPD��� Logaritmos Autoria: Jeanne Dobgenski Como citar esse documento: DOBGENSKI, Jeanne. Cálculo I: Logaritmos. Caderno de Atividades. Valinhos: Anhanguera Educacional, 2014. Índice ������$QKDQJXHUD�(GXFDFLRQDO�� 3URLELGD� D� UHSURGXomR� ¿QDO� RX� SDUFLDO� SRU� TXDOTXHU�PHLR� GH� LPSUHVVmR�� HP� IRUPD� LGrQWLFD�� UHVXPLGD� RX�PRGL¿FDGD� HP� OtQJXD� SRUWXJXHVD�RX�TXDOTXHU�RXWUR�LGLRPD� Pág. 12 Pág. 13 Pág. 14 Pág. 13 Pág. 10Pág. 9 ACOMPANHENAWEB Pág. 3 CONVITEÀLEITURA Pág. 3 PORDENTRODOTEMA � Logaritmos Uma situação comum na qual é possível aplicar logaritmos�p�QD�iUHD�¿QDQFHLUD��9RFr�VDEHULD�D¿UPDU�HP�TXDQWR� WHPSR�XP�FDSLWDO�p�GXSOLFDGR�TXDQGR�DSOLFDGR�D�XPD�WD[D�GH������DR�PrV�HP�MXURV�FRPSRVWRV"�$�UHVSRVWD�FRUUHWD�p�GRLV� DQRV��VHWH�PHVHV�H�YLQWH�H�VHLV�GLDV��7HQWH�UHVROYHU�HVVH�SUREOHPD�XVDQGR�D�IyUPXOD�SDUD�FiOFXOR�GH�juros compostos M=C(1+i)t��VHQGR�0� �PRQWDQWH��&� �FDSLWDO��L� �WD[D�GH�MXURV�H�W�R�WHPSR��9HMD�TXH�QHVVD�VLWXDomR�R�PRQWDQWH�VHUi�R�FDSLWDO� GXSOLFDGR��ORJR�0� ��&��L� �������H�R�WHPSR�VHUi�HQFRQWUDGR�HP�PHVHV��D¿QDO�D�WD[D�GH�MXURV�p�DR�PrV��9RFr�GHYH�WHU� chegado à seguinte situação 2 = 1,022t��0DV��H�DJRUD"�W�p�XP�H[SRHQWH�H�p�SRVVtYHO�SHUFHEHU�TXH�HVVH�H[SRHQWH�GHYH�VHU� R�YDORU�DGHTXDGR�SDUD�WRUQDU�D�EDVH���������LJXDO�D����9RFr�DSUHQGHX�D�FDOFXODU�XP�Q~PHUR�HOHYDGR�D�XP�H[SRHQWH�TXH� YDULD��RX�VHMD��DOWHUD�VHX�YDORU��HP�IXQo}HV�H[SRQHQFLDLV��QmR�p"�(VVH�FiOFXOR�p�IDFLOPHQWH�HIHWXDGR�FRP�R�XVR�GH�XPD� FDOFXODGRUD�FLHQWt¿FD�XVDQGR�D�IXQomR�GH�H[SRHQWH��1R�HQWDQWR��SDUD�HQFRQWUDU�R�YDORU�TXH�R�H[SRHQWH�GHYH�WHU�SDUD�TXH� um determinado resultado ocorra, como no exemplo 2 = 1,022t��R�FiOFXOR�GHYH�VHU�IHLWR�SRU�PHLR�GH�ORJDULWPRV� (VVD�p�DSHQDV�XPD�GDV�VLWXDo}HV�SUiWLFDV�HP�TXH�R�XVR�GH�ORJDULWPRV�p�IXQGDPHQWDO�SDUD�HQFRQWUDU�D�VROXomR�GH�XP� SUREOHPD��(QWmR��p�LPSRUWDQWH�FRPSUHHQGHU�R�TXH�VLJQL¿FD�ORJDULWPR�H�HQWHQGHU�VXDV�SURSULHGDGHV��SRLV�VmR�HODV�TXH� VHUmR�XVDGDV�QD�VROXomR�GH�GLYHUVDV�VLWXDo}HV� 2V� ORJDULWPRV� VmR� H[WUHPDPHQWH� ~WHLV� SDUD� UHVROYHU� SUREOHPDV� TXH� RFRUUHP� HP� VLWXDo}HV� GLYHUVDV� FRPR� QD� HFRQRPLD��SUHYLVmR�GH�HQFKHQWHV�� FUHVFLPHQWR�SRSXODFLRQDO��DEDORV�VtVPLFRV�HQWUH�YiULDV�RXWUDV��(VVH�HVWXGR�HVWi� LQWLPDPHQWH�OLJDGR�DR�TXH�Mi�IRL�YLVWR�QR�WHPD���H���GHVVD�GLVFLSOLQD��SRLV�HVWXGDU�ORJDULWPRV�HQYROYH�FRQFHLWRV�GH�IXQo}HV� H[SRQHQFLDLV�H� IXQo}HV� LQYHUVDV��([LVWHP�PXLWRV�FDVRV�HP�TXH�p�QHFHVViULR�HQFRQWUDU�FUHVFLPHQWR�RX�GHFDLPHQWR� H[SRQHQFLDO��PDV�D�WD[D�QmR�p�GDGD��3DUD�HQFRQWUi�OD��p�SUHFLVR�FRQKHFHU�D�TXDQWLGDGH�HP�GRLV�LQVWDQWHV�GLIHUHQWHV�H� GHSRLV�UHVROYHU�SDUD�D�WD[D�GH�FUHVFLPHQWR�RX�GHFDLPHQWR��0DV�FRPR�ID]HU�LVVR"�/HLD�R�WH[WR��'HSRLV�IDoD�RV�H[HUFtFLRV� SURSRVWRV�H�DFHVVH�R�PDWHULDO�LQGLFDGR�H�TXH�HVWi�GLVSRQtYHO�QD�ZHE��%RQV�HVWXGRV� CONVITEÀLEITURA L it SURSRVWRV�H�DFHVVH�R�PDWHULDO�LQGLFDGR�H�TXH�HVWi�GLVSRQtYHO�QD�ZHE��%RQV�HVWXGRV� PORDENTRODOTEMA � 2EVHUYH�TXH�QR�H[HPSOR�DSUHVHQWDGR�IRUDP�PHQFLRQDGDV�DV�IXQo}HV�H[SRQHQFLDLV��SDUD�UHOHPEUDU��VH�a>1 e D1, a IXQomR�H[SRQHQFLDO�I�[�� �ax p�FUHVFHQWH�RX�GHFUHVFHQWH��H��SRUWDQWR��LQMHWRUD�SHOR�7HVWH�GD�5HWD�+RUL]RQWDO�TXH�IRL�YLVWR� QR�WHPD�DQWHULRU��,VVR�VLJQL¿FD�TXH�D�IXQomR�H[SRQHQFLDO�p�LQYHUVtYHO��RX�LQYHUWtYHO���FHUWR"�(QWmR��H[LVWH�XPD�IXQomR� inversa f-1, chamada função logarítmica com base a denotada por log a � 6H�XVDUPRV�D�IRUPXODomR�GH�IXQomR�LQYHUVD� vista no tema anterior f-1(x) = y ֞ f(y) = x WHP�VH log a x = y ֞ ay� �[� 'HVVD�IRUPD��VH�x> 0, então o resultado de log a [�HTXLYDOH�DR�H[SRHQWH��\��TXH�GHYH�VH�HOHYDU�D�EDVH�a SDUD�REWHU�x. 3RU�H[HPSOR��ORJ 10 ������ ����SRUTXH������ ��������RX�VHMD��EDVH�D ����H[SRHQWH�\� ����H�[� �������p�R�UHVXOWDGR��3DUD� UHIRUoDU��R�ORJDULWPR�p����TXH�p�R�UHVXOWDGR�GR�H[SRHQWH�TXH�DR�HOHYDU�����REWpP�VH������� 2X�VHMD��R�ORJDULWPR�p�FRQVLGHUDGR�FRPR�D�IXQomR�LQYHUVD�GD�IXQomR�H[SRQHQFLDO��YHMD�D�)LJXUD�������PDV��SRGH�VH�GL]HU� R�ORJDULWPR�QDGD�PDLV�p�TXH�R�Q~PHUR�TXH�VHUYH�GH�H[SRHQWH��&RQVHTXHQWHPHQWH��FDOFXODU�R�ORJDULWPR�GH�XP�Q~PHUR� FRQVLVWH�HP�GHVFREULU�TXDO�p�HVWH�Q~PHUR�TXH�VHUYLUi�GH�expoente j�EDVH�SDUD�HQFRQWUDU�XP�YDORU�GDGR� Figura 4.1�±�*Ui¿FR�GD�IXQomR�H[SRQHQFLDO��x�H�VXD�IXQomR�LQYHUVD�ORJ 2 [� )RQWH��)LQQH\�������� PORDENTRODOTEMA � 'H�IRUPD�JHQpULFD�IRL�XVDGR�R�WHUPR�³D´�SDUD�LQGLFDU�D�EDVH�GR�ORJDULWPR��1D�)LJXUD�����p�DSUHVHQWDGR�R�JUi¿FR�SDUD� D�EDVH����SRLV�D�IXQomR�H[SRQHQFLDO�WHP�EDVH�����QR�H[HPSOR�QXPpULFR�ORJ 10 ������� ����D�EDVH�p�����TXH�p�XPD�EDVH� EDVWDQWH�XVXDO�HP�ORJDULWPRV��WDQWR�TXH�PXLWDV�YH]HV�D�EDVH�p�RPLWLGD��RX�VHMD��ORJ 10 ������ �ORJ������� �����$�)LJXUD����� DSUHVHQWD�R�JUi¿FR�GD�IXQomR�H[SRQHQFLDO�H�ORJDUtWPLFD�GH�EDVH����� Figura 4.2�±�2�JUi¿FR�GH�y = 10x e y = logx. )RQWH��+XJKHV�+DOOHWW��������S������ 3RUWDQWR��R�ORJDULWPR�GH�x HP�EDVH�����HVFULWR�ORJ 10 x, p�D�SRWrQFLD�GH����TXH�p�QHFHVViULD�SDUD�REWHU�x. (P�RXWUDV� palavras, log10x �F�VLJQL¿FD�TXH��� c= x,�VHQGR�HVFULWR��PXLWDV�YH]HV��ORJ�x HP�YH]�GH�log10x. $V�SURSULHGDGHV�GRV�DOJRULWPRV�VHUmR�GHVFULWDV�HP�WHUPRV�GH�XPD�EDVH�JHQpULFD�³D´��OHPEUDQGR�TXH�D�EDVH�³D´�SRGHUi� VHU�FRQVLGHUDGD�FRPR�TXDOTXHU�YDORU��'HVVD� IRUPD��DV�HTXDo}HV�GH�FDQFHODPHQWR�YLVWDV�QR� WHPD�DQWHULRU��TXDQGR� aplicadas a f(x) = ax e f-1(x) = log a x, podem ser escritas em termos de logaritmos como log a (ax)=x para todo x , e alog a x� �[�SDUD�WRGR�[�!��� $�IXQomR�ORJDUtWPLFD�ORJ a WHP�R�GRPtQLR������H�D�LPDJHP���6HX�JUi¿FR�p�D�UHÀH[mR�GR�JUi¿FR�GH�y = ax em torno da reta y = x��FRPR�IRL�YLVWR�QDV�)LJXUDV�����H����� PORDENTRODOTEMA � 2EVHUYH�TXH�D� IXQomR�H[SRQHQFLDO� WHP�XP�FUHVFLPHQWR�PXLWR� UiSLGR�TXDQGR�[�!���H��FRQVHTXHQWHPHQWH��D� IXQomR� ORJDUtWPLFD� WHP� XP� FUHVFLPHQWR�PXLWR� OHQWR� SDUD� [� !���0DV� SRUTXH� D� SDUWLU� GH� [� �"�&RPR� QmR� Ki� SRWrQFLD� FXMR� UHWRUQR�VHMD�]HUR��HQWmR�D�IXQomR�ORJDUtWPLFD�QmR�HVWi�GH¿QLGD�SDUD�HVVH�YDORU�H�SRU�LVVR�WRGRV�RV�JUi¿FRV�GDV�IXQo}HV� ORJDUtWPLFDV�SDVVDP�SHOR�SRQWR�������±�YHU�)LJXUD����� Figura 4.3�±�)XQo}HV�ORJDUtWPLFDV�GH�YiULDV�EDVHV� )RQWH��6WHZDUW��������S������ Propriedades dos logaritmos: se x e y IRUHP�Q~PHURV�SRVLWLYRV��HQWmR� to).cancelamen de função segunda(0a6) to)cancelamen de função primeira()(log 5) base de mudança)(log )(log)(log4) real númeroqualquer r sendo)(log)(log)3 )(log)(loglog)2 )(log)(log)(log)1 )(log !� � � ¸¸¹ · ¨¨© § � xx xxa b aa xrx yx y x yxxy x x a c c b a r a aaa aaa a $JRUD�YRFr�SRVVXL�DV�IHUUDPHQWDV�QHFHVViULDV�SDUD�UHVROYHU�R�H[HPSOR�LQLFLDO�GHVVH�WHPD� PORDENTRODOTEMA � Exemplo 1�±�(P�TXDQWR�WHPSR�XP�FDSLWDO�p�GXSOLFDGR�TXDQGR�VXEPHWLGR�j�XPD�DSOLFDomR�GH�MXURV�FRPSRVWRV�FRP�WD[D� GH������DR�PrV" Solução��XVDQGR�D�IyUPXOD�SDUD�MXURV�FRPSRVWRV�0 &���L�t��WHP�VH 0� ��&��R�PRQWDQWH�VHUi�GXDV�YH]HV�R�FDSLWDO��&�� L� ��������p�D�WD[D�GH�MXURV� W� �"��p�D�LQFyJQLWD�H�TXH�VH�GHVHMD�GHVFREULU�±�R�WHPSR�SDUD�TXH�D�DSOLFDomR�GXSOLTXH� M=C(1+i)t 2C = C(1+0,022)t 2C = C(1,022)t 2 = 1,022t 1HVVH�SRQWR�p�SRVVtYHO�DSOLFDU�DV�SURSULHGDGHV�GH�ORJDULWPRV�H�Ki�GXDV�IRUPDV�GH�UHVROYHU�� ���0XGDQoD�GH�EDVH�±�SHOD�GH¿QLomR�GH�ORJDULWPRV�ORJ a x = y ֞ ay� �[��WHP�VH��D ��������[� ���H�\� �W��3RUWDQWR� meses. 31,85 )científica acalculador pela 10 base na logs esses calculando(022,1log 2log2log 022,1 t tt W� ����PHVHV�H����GLDV�RX���DQRV����PHVHV�H����GLDV� 2 = 1,022t log2 = log1,022t ���$SOLFDU�ORJ�QRV�GRLV�ODGRV�GD�HTXDomR�±�VHPSUH�p�SRVVtYHO�UHVROYHU�XPD�HTXDomR�HIHWXDQGR�D�PHVPD�RSHUDomR�HP� DPERV�ODGRV�GD�LJXDOGDGH��FHUWR"�/RJR� 2 = 1,022t log2 = log1,022t�H�SHOD�SURSULHGDGH������SRGH�VH�HVFUHYHU ORJ�� �W�ORJ������TXH�FKHJDUi�QD�PHVPD�GLYLVmR�GD�VROXomR�DQWHULRU��SRUWDQWR��W� �������PHVHV�RX���DQRV����PHVHV�H����GLDV� $�RXWUD�EDVH�PXLWR�XWLOL]DGD�p�e. 2�ORJDULWPR�HP�EDVH�e é chamado de logaritmo natural de x, denotado por ln x e GH¿QLGR�FRPR�VHQGR�D�IXQomR�LQYHUVD�GH�ex, RX�VHMD��R�ORJDULWPR�QDWXUDO�GH�x, escrito ln x, p�D�SRWrQFLD�GH�e necessária SDUD�REWHU�x. 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