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 \(y=14+2⋅{}^{4}\!\log(8x-1)\) Vrijmaken 00kn - Logaritmische formules herleiden - basis - 2ms - dynamic variables ○ \(y=14+2⋅{}^{4}\!\log(8x-1)\) \(2⋅{}^{4}\!\log(8x-1)=y-14\) \({}^{4}\!\log(8x-1)=\frac{1}{2}y-7\) 1p ○ \(8x-1=4^{\frac{1}{2}y-7}\) 1p ○ \(8x=4^{\frac{1}{2}y-7}+1\) \(x=\frac{1}{8}⋅4^{\frac{1}{2}y-7}+\frac{1}{8}\) 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 \(y=2{,}93⋅{}^{3}\!\log(x)+2{,}35\) in de vorm \(y={}^{3}\!\log(ax^b)\text{.}\) Geef \(a\) en \(b\) in twee decimalen. Herleiden (4) 00l0 - Logaritmische formules herleiden - basis - 0ms - dynamic variables a \(y=2{,}93⋅{}^{3}\!\log(x)+2{,}35\) \(\text{ }={}^{3}\!\log(x^{2{,}93})+2{,}35\) 1p ○ \(\text{ }={}^{3}\!\log(x^{2{,}93})+{}^{3}\!\log(3^{2{,}35})\) \(\text{ }={}^{3}\!\log(x^{2{,}93}⋅3^{2{,}35})\) 1p ○ \(\text{ }={}^{3}\!\log(x^{2{,}93}⋅13{,}220...)\) Dus \(y={}^{3}\!\log(13{,}22⋅x^{2{,}93})\text{.}\) 1p 3p b Schrijf de formule \(y={}^{3}\!\log(37x^4\sqrt{x})\) in de vorm \(y=a+b⋅{}^{3}\!\log(x)\text{.}\) Geef \(a\) in twee decimalen. Logaritmisch (5) 00l1 - Logaritmische formules herleiden - basis - 1ms - dynamic variables b \(y={}^{3}\!\log(37x^4\sqrt{x})\) \(\text{ }={}^{3}\!\log(37x^{4{,}5})\) 1p ○ \(\text{ }={}^{3}\!\log(37)+{}^{3}\!\log(x^{4{,}5})\) \(\text{ }={}^{3}\!\log(37)+4{,}5⋅{}^{3}\!\log(x)\) 1p ○ \(\text{ }=3{,}286...+4{,}5⋅{}^{3}\!\log(x)\) Dus \(y=3{,}29+4{,}5⋅{}^{3}\!\log(x)\text{.}\) 1p 3p c Schrijf de formule \(y={}^{2}\!\log(1{,}2x)-2{,}6\) in de vorm \(y=a+b⋅{}^{4}\!\log(x)\text{.}\) Geef \(a\) en \(b\) in twee decimalen. Herleiden (6) 00l2 - Logaritmische formules herleiden - basis - 1ms - dynamic variables c \(y={}^{2}\!\log(1{,}2x)-2{,}6\) \(\text{ }={}^{2}\!\log(1{,}2)+{}^{2}\!\log(x)-2{,}6\) 1p ○ \(\text{ }={}^{2}\!\log(1{,}2)-2{,}6+{{}^{4}\!\log(x) \over {}^{4}\!\log(2)}\) \(\text{ }={}^{2}\!\log(1{,}2)-2{,}6+{1 \over {}^{4}\!\log(2)}⋅{}^{4}\!\log(x)\) 1p ○ \(\text{ }=0{,}263...-2{,}6+{1 \over 0{,}5}⋅{}^{4}\!\log(x)\) \(\text{ }=-2{,}336...+2⋅{}^{4}\!\log(x)\) Dus \(y=-2{,}34+2{,}00⋅{}^{4}\!\log(x)\text{.}\) 1p 3p d Schrijf de formule \(y=6⋅{}^{2}\!\log(32x)-7\) in de vorm \(y=a+b⋅{}^{2}\!\log(4x)\text{.}\) Herleiden (7) 00l3 - Logaritmische formules herleiden - basis - 1ms - dynamic variables d \(y=6⋅{}^{2}\!\log(32x)-7\) \(\text{ }=6⋅({}^{2}\!\log(8)+{}^{2}\!\log(4x))-7\) 1p ○ \(\text{ }=6⋅(3+{}^{2}\!\log(4x))-7\) 1p ○ \(\text{ }=18+6⋅{}^{2}\!\log(4x)-7\) \(\text{ }=11+6⋅{}^{2}\!\log(4x)\) 1p
havo wiskunde B	9.4 Formules omwerken
Logaritmische formules herleiden (6)	opgave 1 Herleid tot de gevraagde vorm. 3p a Schrijf de formule \(y=9\,600⋅0{,}92^x\) 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 \(y=9\,600⋅0{,}92^x\) \(\log(y)=\log(9\,600⋅0{,}92^x)\) \(\log(y)=\log(9\,600)+\log(0{,}92^x)\) 1p ○ \(\log(y)=\log(9\,600)+x⋅\log(0{,}92)\) 1p ○ \(\log(y)=3{,}982...+x⋅-0{,}03621...\) Dus \(\log(y)=-0{,}0362x+3{,}98\) 1p 3p b Schrijf de formule \(y=1\,000⋅0{,}94^{6x+2}\) 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 \(y=1\,000⋅0{,}94^{6x+2}\) \(\log(y)=\log(1\,000⋅0{,}94^{6x+2})\) \(\log(y)=\log(1\,000)+\log(0{,}94^{6x+2})\) 1p ○ \(\log(y)=\log(1\,000)+(6x+2)⋅\log(0{,}94)\) \(\log(y)=\log(1\,000)+6x⋅\log(0{,}94)+2⋅\log(0{,}94)\) 1p ○ \(\log(y)=3+6x⋅-0{,}02687...+2⋅-0{,}02687...\) \(\log(y)=3-0{,}16123...⋅x-0{,}05374...\) Dus \(\log(y)=-0{,}1612x+2{,}95\) 1p 3p c Schrijf de formule \(\log(y)=-0{,}4791x+3{,}56\) 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 \(\log(y)=-0{,}4791x+3{,}56\) \(y=10^{-0{,}4791x+3{,}56}\) 1p ○ \(y=10^{-0{,}4791x}⋅10^{3{,}56}\) \(y=(10^{-0{,}4791})^x⋅10^{3{,}56}\) 1p ○ \(y=0{,}331...^x⋅3630{,}780...\) Dus \(y=3\,631⋅0{,}33^x\text{.}\) 1p 3p d Schrijf de formule \(\log(y)=1{,}84+1{,}85⋅\log(x)\) in de vorm \(y=ax^b\text{.}\) Geef \(a\) in gehelen. Dubbel (3) 00kr - Logaritmische formules herleiden - basis - 0ms - dynamic variables d \(\log(y)=1{,}84+1{,}85⋅\log(x)\) \(\log(y)=\log(10^{1{,}84})+\log(x^{1{,}85})\) \(\log(y)=\log(10^{1{,}84}⋅x^{1{,}85})\) 1p ○ \(y=10^{1{,}84}⋅x^{1{,}85}\) 1p ○ \(y=69{,}183...⋅x^{1{,}85}\) Dus \(y=69⋅x^{1{,}85}\text{.}\) 1p opgave 2 Herleid tot de gevraagde vorm. 3p a Schrijf de formule \(y=390x^{1{,}52}\) 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 \(y=390x^{1{,}52}\) \(\log(y)=\log(390x^{1{,}52})\) 1p ○ \(\log(y)=\log(390)+\log(x^{1{,}52})\) \(\log(y)=\log(390)+1{,}52⋅\log(x)\) 1p ○ \(\log(y)=2{,}591...+1{,}52⋅\log(x)\) Dus \(y=2{,}59+1{,}52⋅\log(x)\text{.}\) 1p 3p b Schrijf de formule \(y={390 \over x^4}\) 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 \(y={390 \over x^4}=390x^{-4}\) \(\log(y)=\log(390x^{-4})\) 1p ○ \(\log(y)=\log(390)+\log(x^{-4})\) \(\log(y)=\log(390)-4⋅\log(x)\) 1p ○ \(\log(y)=2{,}591...-4⋅\log(x)\) Dus \(y=2{,}59-4⋅\log(x)\text{.}\) 1p

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