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 \(y = 36 + 4 ⋅ {}^{2}\!\log(8 x - 7)\) Vrijmaken 00kn - Logaritmische formules herleiden - basis - 0ms - dynamic variables ○ \(y = 36 + 4 ⋅ {}^{2}\!\log(8 x - 7)\) \(4 ⋅ {}^{2}\!\log(8 x - 7) = y - 36\) \({}^{2}\!\log(8 x - 7) = \frac{1}{4} y - 9\) 1p ○ \(8 x - 7 = 2^{\frac{1}{4} y - 9}\) 1p ○ \(8 x = 2^{\frac{1}{4} y - 9} + 7\) \(x = \frac{1}{8} ⋅ 2^{\frac{1}{4} y - 9} + \frac{7}{8}\) 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 \(y = 1{,}69 ⋅ {}^{2}\!\log(x) + 2{,}86\) in de vorm \(y = {}^{2}\!\log(a x^{b}) \text{.}\) Geef \(a\) en \(b\) in twee decimalen. Herleiden (4) 00l0 - Logaritmische formules herleiden - basis - 0ms - dynamic variables a \(y = 1{,}69 ⋅ {}^{2}\!\log(x) + 2{,}86\) \(\text{ } = {}^{2}\!\log(x^{1{,}69}) + 2{,}86\) 1p ○ \(\text{ } = {}^{2}\!\log(x^{1{,}69}) + {}^{2}\!\log(2^{2{,}86})\) \(\text{ } = {}^{2}\!\log(x^{1{,}69} ⋅ 2^{2{,}86})\) 1p ○ \(\text{ } = {}^{2}\!\log(x^{1{,}69} ⋅ 7{,}260...)\) Dus \(y = {}^{2}\!\log(7{,}26 ⋅ x^{1{,}69}) \text{.}\) 1p 3p b Schrijf de formule \(y = {}^{5}\!\log({42 \over x^{4} \sqrt{x}})\) in de vorm \(y = a + b ⋅ {}^{5}\!\log(x) \text{.}\) Geef \(a\) in twee decimalen. Herleiden (5) 00l1 - Logaritmische formules herleiden - basis - 0ms - dynamic variables b \(y = {}^{5}\!\log({42 \over x^{4} \sqrt{x}})\) \(\text{ } = {}^{5}\!\log(42 x^{-4{,}5})\) 1p ○ \(\text{ } = {}^{5}\!\log(42) + {}^{5}\!\log(x^{-4{,}5})\) \(\text{ } = {}^{5}\!\log(42) - 4{,}5 ⋅ {}^{5}\!\log(x)\) 1p ○ \(\text{ } = 2{,}322... - 4{,}5 ⋅ {}^{5}\!\log(x)\) Dus \(y = 2{,}32 - 4{,}5 ⋅ {}^{5}\!\log(x) \text{.}\) 1p 3p c Schrijf de formule \(y = {}^{3}\!\log(1{,}9 x) - 2{,}9\) in de vorm \(y = a + b ⋅ {}^{2}\!\log(x) \text{.}\) Geef \(a\) en \(b\) in twee decimalen. Herleiden (6) 00l2 - Logaritmische formules herleiden - basis - 0ms - dynamic variables c \(y = {}^{3}\!\log(1{,}9 x) - 2{,}9\) \(\text{ } = {}^{3}\!\log(1{,}9) + {}^{3}\!\log(x) - 2{,}9\) 1p ○ \(\text{ } = {}^{3}\!\log(1{,}9) - 2{,}9 + {{}^{2}\!\log(x) \over {}^{2}\!\log(3)}\) \(\text{ } = {}^{3}\!\log(1{,}9) - 2{,}9 + {1 \over {}^{2}\!\log(3)} ⋅ {}^{2}\!\log(x)\) 1p ○ \(\text{ } = 0{,}584... - 2{,}9 + {1 \over 1{,}584...} ⋅ {}^{2}\!\log(x)\) \(\text{ } = -2{,}315... + 0{,}630... ⋅ {}^{2}\!\log(x)\) Dus \(y = -2{,}32 + 0{,}63 ⋅ {}^{2}\!\log(x) \text{.}\) 1p 3p d Schrijf de formule \(y = 10 ⋅ {}^{2}\!\log(48 x) - 8\) in de vorm \(y = a + b ⋅ {}^{2}\!\log(3 x) \text{.}\) Herleiden (7) 00l3 - Logaritmische formules herleiden - basis - 0ms - dynamic variables d \(y = 10 ⋅ {}^{2}\!\log(48 x) - 8\) \(\text{ } = 10 ⋅ ({}^{2}\!\log(16) + {}^{2}\!\log(3 x)) - 8\) 1p ○ \(\text{ } = 10 ⋅ (4 + {}^{2}\!\log(3 x)) - 8\) 1p ○ \(\text{ } = 40 + 10 ⋅ {}^{2}\!\log(3 x) - 8\) \(\text{ } = 32 + 10 ⋅ {}^{2}\!\log(3 x)\) 1p
havo wiskunde B	9.4 Formules omwerken
Logaritmische formules herleiden (6)	opgave 1 Herleid tot de gevraagde vorm. 3p a Schrijf de formule \(y = 4\,100 ⋅ 0{,}92^{x}\) in de vorm \(\log(y) = a x + b \text{.}\) Geef \(a\) in vier decimalen en \(b\) in twee decimalen. Herleiden (1) 00ko - Logaritmische formules herleiden - basis - 0ms - dynamic variables a \(y = 4\,100 ⋅ 0{,}92^{x}\) \(\log(y) = \log(4\,100 ⋅ 0{,}92^{x})\) \(\log(y) = \log(4\,100) + \log(0{,}92^{x})\) 1p ○ \(\log(y) = \log(4\,100) + x ⋅ \log(0{,}92)\) 1p ○ \(\log(y) = 3{,}612... + x ⋅ -0{,}03621...\) Dus \(\log(y) = -0{,}0362 x + 3{,}61\) 1p 3p b Schrijf de formule \(y = 7\,100 ⋅ 1{,}17^{4 x + 3}\) in de vorm \(\log(y) = a x + b \text{.}\) Geef \(a\) in vier decimalen en \(b\) in twee decimalen. Herleiden (2) 00kp - Logaritmische formules herleiden - basis - 0ms - dynamic variables b \(y = 7\,100 ⋅ 1{,}17^{4 x + 3}\) \(\log(y) = \log(7\,100 ⋅ 1{,}17^{4 x + 3})\) \(\log(y) = \log(7\,100) + \log(1{,}17^{4 x + 3})\) 1p ○ \(\log(y) = \log(7\,100) + (4 x + 3) ⋅ \log(1{,}17)\) \(\log(y) = \log(7\,100) + 4 x ⋅ \log(1{,}17) + 3 ⋅ \log(1{,}17)\) 1p ○ \(\log(y) = 3{,}851... + 4 x ⋅ 0{,}06818... + 3 ⋅ 0{,}06818...\) \(\log(y) = 3{,}851... + 0{,}27274... ⋅ x + 0{,}20455...\) Dus \(\log(y) = 0{,}2727 x + 4{,}06\) 1p 3p c Schrijf de formule \(\log(y) = -0{,}3243 x + 2{,}49\) 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 \(\log(y) = -0{,}3243 x + 2{,}49\) \(y = 10^{-0{,}3243 x + 2{,}49}\) 1p ○ \(y = 10^{-0{,}3243 x} ⋅ 10^{2{,}49}\) \(y = (10^{-0{,}3243})^{x} ⋅ 10^{2{,}49}\) 1p ○ \(y = 0{,}473...^{x} ⋅ 309{,}029...\) Dus \(y = 309 ⋅ 0{,}47^{x} \text{.}\) 1p 3p d Schrijf de formule \(\log(y) = 1{,}19 - 1{,}01 ⋅ \log(x)\) in de vorm \(y = a x^{b} \text{.}\) Geef \(a\) in gehelen. Dubbel (3) 00kr - Logaritmische formules herleiden - basis - 0ms - dynamic variables d \(\log(y) = 1{,}19 - 1{,}01 ⋅ \log(x)\) \(\log(y) = \log(10^{1{,}19}) + \log(x^{-1{,}01})\) \(\log(y) = \log(10^{1{,}19} ⋅ x^{-1{,}01})\) 1p ○ \(y = 10^{1{,}19} ⋅ x^{-1{,}01}\) 1p ○ \(y = 15{,}488... ⋅ x^{-1{,}01}\) Dus \(y = 15 ⋅ x^{-1{,}01} \text{.}\) 1p opgave 2 Herleid tot de gevraagde vorm. 3p a Schrijf de formule \(y = 480 x^{1{,}98}\) 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 \(y = 480 x^{1{,}98}\) \(\log(y) = \log(480 x^{1{,}98})\) 1p ○ \(\log(y) = \log(480) + \log(x^{1{,}98})\) \(\log(y) = \log(480) + 1{,}98 ⋅ \log(x)\) 1p ○ \(\log(y) = 2{,}681... + 1{,}98 ⋅ \log(x)\) Dus \(y = 2{,}68 + 1{,}98 ⋅ \log(x) \text{.}\) 1p 3p b Schrijf de formule \(y = {600 \over x^{2} \sqrt{x}}\) 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 \(y = {600 \over x^{2} \sqrt{x}} = 600 x^{-2{,}5}\) \(\log(y) = \log(600 x^{-2{,}5})\) 1p ○ \(\log(y) = \log(600) + \log(x^{-2{,}5})\) \(\log(y) = \log(600) - 2{,}5 ⋅ \log(x)\) 1p ○ \(\log(y) = 2{,}778... - 2{,}5 ⋅ \log(x)\) Dus \(y = 2{,}78 - 2{,}5 ⋅ \log(x) \text{.}\) 1p

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