Getal & Ruimte (12e editie) - havo wiskunde B

'Logaritmische formules herleiden'.

havo wiskunde B	9.2 Werken met logaritmen
Logaritmische formules herleiden (1)	opgave 1 Druk \(x\) uit in \(y\text{.}\) 3p \(y=6+2⋅{}^{6}\!\log(9x+5)\) Vrijmaken 00kn - Logaritmische formules herleiden - basis - 2ms - dynamic variables ○ \(y=6+2⋅{}^{6}\!\log(9x+5)\) \(2⋅{}^{6}\!\log(9x+5)=y-6\) \({}^{6}\!\log(9x+5)=\frac{1}{2}y-3\) 1p ○ \(9x+5=6^{\frac{1}{2}y-3}\) 1p ○ \(9x=6^{\frac{1}{2}y-3}-5\) \(x=\frac{1}{9}⋅6^{\frac{1}{2}y-3}-\frac{5}{9}\) 1p
havo wiskunde B	9.3 Rekenregels voor logaritmen
Logaritmische formules herleiden (4)	opgave 1 Herleid tot de gevraagde vorm. 3p a Schrijf de formule \(y=1{,}41⋅{}^{2}\!\log(x)-2{,}92\) in de vorm \(y={}^{2}\!\log(ax^b)\text{.}\) Geef \(a\) en \(b\) in twee decimalen. Herleiden (4) 00l0 - Logaritmische formules herleiden - basis - 0ms - dynamic variables a \(y=1{,}41⋅{}^{2}\!\log(x)-2{,}92\) \(\text{ }={}^{2}\!\log(x^{1{,}41})-2{,}92\) 1p ○ \(\text{ }={}^{2}\!\log(x^{1{,}41})+{}^{2}\!\log(2^{-2{,}92})\) \(\text{ }={}^{2}\!\log(x^{1{,}41}⋅2^{-2{,}92})\) 1p ○ \(\text{ }={}^{2}\!\log(x^{1{,}41}⋅0{,}132...)\) Dus \(y={}^{2}\!\log(0{,}13⋅x^{1{,}41})\text{.}\) 1p 3p b Schrijf de formule \(y={}^{4}\!\log({77 \over x^5})\) in de vorm \(y=a+b⋅{}^{4}\!\log(x)\text{.}\) Geef \(a\) in twee decimalen. Logaritmisch (5) 00l1 - Logaritmische formules herleiden - basis - 1ms - dynamic variables b \(y={}^{4}\!\log({77 \over x^5})\) \(\text{ }={}^{4}\!\log(77x^{-5})\) 1p ○ \(\text{ }={}^{4}\!\log(77)+{}^{4}\!\log(x^{-5})\) \(\text{ }={}^{4}\!\log(77)-5⋅{}^{4}\!\log(x)\) 1p ○ \(\text{ }=3{,}133...-5⋅{}^{4}\!\log(x)\) Dus \(y=3{,}13-5⋅{}^{4}\!\log(x)\text{.}\) 1p 3p c Schrijf de formule \(y={}^{5}\!\log(2{,}6x)+1{,}6\) in de vorm \(y=a+b⋅{}^{2}\!\log(x)\text{.}\) Geef \(a\) en \(b\) in twee decimalen. Herleiden (6) 00l2 - Logaritmische formules herleiden - basis - 1ms - dynamic variables c \(y={}^{5}\!\log(2{,}6x)+1{,}6\) \(\text{ }={}^{5}\!\log(2{,}6)+{}^{5}\!\log(x)+1{,}6\) 1p ○ \(\text{ }={}^{5}\!\log(2{,}6)+1{,}6+{{}^{2}\!\log(x) \over {}^{2}\!\log(5)}\) \(\text{ }={}^{5}\!\log(2{,}6)+1{,}6+{1 \over {}^{2}\!\log(5)}⋅{}^{2}\!\log(x)\) 1p ○ \(\text{ }=0{,}593...+1{,}6+{1 \over 2{,}321...}⋅{}^{2}\!\log(x)\) \(\text{ }=2{,}193...+0{,}430...⋅{}^{2}\!\log(x)\) Dus \(y=2{,}19+0{,}43⋅{}^{2}\!\log(x)\text{.}\) 1p 3p d Schrijf de formule \(y=8⋅{}^{4}\!\log(48x)-9\) in de vorm \(y=a+b⋅{}^{4}\!\log(3x)\text{.}\) Herleiden (7) 00l3 - Logaritmische formules herleiden - basis - 1ms - dynamic variables d \(y=8⋅{}^{4}\!\log(48x)-9\) \(\text{ }=8⋅({}^{4}\!\log(16)+{}^{4}\!\log(3x))-9\) 1p ○ \(\text{ }=8⋅(2+{}^{4}\!\log(3x))-9\) 1p ○ \(\text{ }=16+8⋅{}^{4}\!\log(3x)-9\) \(\text{ }=7+8⋅{}^{4}\!\log(3x)\) 1p
havo wiskunde B	9.4 Formules omwerken
Logaritmische formules herleiden (6)	opgave 1 Herleid tot de gevraagde vorm. 3p a Schrijf de formule \(y=5\,500⋅1{,}19^x\) in de vorm \(\log(y)=ax+b\text{.}\) Geef \(a\) in vier decimalen en \(b\) in twee decimalen. Herleiden (1) 00ko - Logaritmische formules herleiden - basis - 1ms - dynamic variables a \(y=5\,500⋅1{,}19^x\) \(\log(y)=\log(5\,500⋅1{,}19^x)\) \(\log(y)=\log(5\,500)+\log(1{,}19^x)\) 1p ○ \(\log(y)=\log(5\,500)+x⋅\log(1{,}19)\) 1p ○ \(\log(y)=3{,}740...+x⋅0{,}07554...\) Dus \(\log(y)=0{,}0755x+3{,}74\) 1p 3p b Schrijf de formule \(y=1\,200⋅0{,}88^{2x+3}\) in de vorm \(\log(y)=ax+b\text{.}\) Geef \(a\) in vier decimalen en \(b\) in twee decimalen. Herleiden (2) 00kp - Logaritmische formules herleiden - basis - 1ms - dynamic variables b \(y=1\,200⋅0{,}88^{2x+3}\) \(\log(y)=\log(1\,200⋅0{,}88^{2x+3})\) \(\log(y)=\log(1\,200)+\log(0{,}88^{2x+3})\) 1p ○ \(\log(y)=\log(1\,200)+(2x+3)⋅\log(0{,}88)\) \(\log(y)=\log(1\,200)+2x⋅\log(0{,}88)+3⋅\log(0{,}88)\) 1p ○ \(\log(y)=3{,}079...+2x⋅-0{,}05551...+3⋅-0{,}05551...\) \(\log(y)=3{,}079...-0{,}11103...⋅x-0{,}16655...\) Dus \(\log(y)=-0{,}1110x+2{,}91\) 1p 3p c Schrijf de formule \(\log(y)=0{,}0402x+2{,}95\) in de vorm \(y=b⋅g^x\text{.}\) Geef \(b\) in gehelen en \(g\) in twee decimalen. Herleiden (3) 00kq - Logaritmische formules herleiden - basis - 0ms - dynamic variables c \(\log(y)=0{,}0402x+2{,}95\) \(y=10^{0{,}0402x+2{,}95}\) 1p ○ \(y=10^{0{,}0402x}⋅10^{2{,}95}\) \(y=(10^{0{,}0402})^x⋅10^{2{,}95}\) 1p ○ \(y=1{,}096...^x⋅891{,}250...\) Dus \(y=891⋅1{,}10^x\text{.}\) 1p 3p d Schrijf de formule \(\log(y)=2{,}66-1{,}89⋅\log(x)\) in de vorm \(y=ax^b\text{.}\) Geef \(a\) in gehelen. Dubbel (3) 00kr - Logaritmische formules herleiden - basis - 0ms - dynamic variables d \(\log(y)=2{,}66-1{,}89⋅\log(x)\) \(\log(y)=\log(10^{2{,}66})+\log(x^{-1{,}89})\) \(\log(y)=\log(10^{2{,}66}⋅x^{-1{,}89})\) 1p ○ \(y=10^{2{,}66}⋅x^{-1{,}89}\) 1p ○ \(y=457{,}088...⋅x^{-1{,}89}\) Dus \(y=457⋅x^{-1{,}89}\text{.}\) 1p opgave 2 Herleid tot de gevraagde vorm. 3p a Schrijf de formule \(y=330x^{1{,}05}\) in de vorm \(\log(y)=a+b⋅\log(x)\text{.}\) Geef \(a\) in twee decimalen. Dubbel (1) 00ks - Logaritmische formules herleiden - basis - 0ms - dynamic variables a \(y=330x^{1{,}05}\) \(\log(y)=\log(330x^{1{,}05})\) 1p ○ \(\log(y)=\log(330)+\log(x^{1{,}05})\) \(\log(y)=\log(330)+1{,}05⋅\log(x)\) 1p ○ \(\log(y)=2{,}518...+1{,}05⋅\log(x)\) Dus \(y=2{,}52+1{,}05⋅\log(x)\text{.}\) 1p 3p b Schrijf de formule \(y={80 \over x^2\sqrt{x}}\) in de vorm \(\log(y)=a+b⋅\log(x)\text{.}\) Geef \(a\) in twee decimalen. Dubbel (2) 00kt - Logaritmische formules herleiden - basis - 0ms - dynamic variables b \(y={80 \over x^2\sqrt{x}}=80x^{-2{,}5}\) \(\log(y)=\log(80x^{-2{,}5})\) 1p ○ \(\log(y)=\log(80)+\log(x^{-2{,}5})\) \(\log(y)=\log(80)-2{,}5⋅\log(x)\) 1p ○ \(\log(y)=1{,}903...-2{,}5⋅\log(x)\) Dus \(y=1{,}90-2{,}5⋅\log(x)\text{.}\) 1p

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