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 \(y = 12 + 4 ⋅ {}^{6}\!\log(9 x - 7)\) Vrijmaken 00kn - Logaritmische formules herleiden - basis - 0ms - dynamic variables ○ \(y = 12 + 4 ⋅ {}^{6}\!\log(9 x - 7)\) \(4 ⋅ {}^{6}\!\log(9 x - 7) = y - 12\) \({}^{6}\!\log(9 x - 7) = \frac{1}{4} y - 3\) 1p ○ \(9 x - 7 = 6^{\frac{1}{4} y - 3}\) 1p ○ \(9 x = 6^{\frac{1}{4} y - 3} + 7\) \(x = \frac{1}{9} ⋅ 6^{\frac{1}{4} y - 3} + \frac{7}{9}\) 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 \(y = 2{,}65 ⋅ {}^{4}\!\log(x) - 2{,}55\) in de vorm \(y = {}^{4}\!\log(a x^{b}) \text{.}\) Geef \(a\) en \(b\) in twee decimalen. Herleiden (4) 00l0 - Logaritmische formules herleiden - basis - 0ms - dynamic variables a \(y = 2{,}65 ⋅ {}^{4}\!\log(x) - 2{,}55\) \(\text{ } = {}^{4}\!\log(x^{2{,}65}) - 2{,}55\) 1p ○ \(\text{ } = {}^{4}\!\log(x^{2{,}65}) + {}^{4}\!\log(4^{-2{,}55})\) \(\text{ } = {}^{4}\!\log(x^{2{,}65} ⋅ 4^{-2{,}55})\) 1p ○ \(\text{ } = {}^{4}\!\log(x^{2{,}65} ⋅ 0{,}029...)\) Dus \(y = {}^{4}\!\log(0{,}03 ⋅ x^{2{,}65}) \text{.}\) 1p 3p b Schrijf de formule \(y = {}^{3}\!\log(53 x^{4})\) in de vorm \(y = a + b ⋅ {}^{3}\!\log(x) \text{.}\) Geef \(a\) in twee decimalen. Herleiden (5) 00l1 - Logaritmische formules herleiden - basis - 0ms - dynamic variables b \(y = {}^{3}\!\log(53 x^{4})\) \(\text{ } = {}^{3}\!\log(53 x^{4})\) 1p ○ \(\text{ } = {}^{3}\!\log(53) + {}^{3}\!\log(x^{4})\) \(\text{ } = {}^{3}\!\log(53) + 4 ⋅ {}^{3}\!\log(x)\) 1p ○ \(\text{ } = 3{,}613... + 4 ⋅ {}^{3}\!\log(x)\) Dus \(y = 3{,}61 + 4 ⋅ {}^{3}\!\log(x) \text{.}\) 1p 3p c Schrijf de formule \(y = {}^{5}\!\log(1{,}2 x) + 0{,}6\) in de vorm \(y = a + b ⋅ {}^{3}\!\log(x) \text{.}\) Geef \(a\) en \(b\) in twee decimalen. Herleiden (6) 00l2 - Logaritmische formules herleiden - basis - 0ms - dynamic variables c \(y = {}^{5}\!\log(1{,}2 x) + 0{,}6\) \(\text{ } = {}^{5}\!\log(1{,}2) + {}^{5}\!\log(x) + 0{,}6\) 1p ○ \(\text{ } = {}^{5}\!\log(1{,}2) + 0{,}6 + {{}^{3}\!\log(x) \over {}^{3}\!\log(5)}\) \(\text{ } = {}^{5}\!\log(1{,}2) + 0{,}6 + {1 \over {}^{3}\!\log(5)} ⋅ {}^{3}\!\log(x)\) 1p ○ \(\text{ } = 0{,}113... + 0{,}6 + {1 \over 1{,}464...} ⋅ {}^{3}\!\log(x)\) \(\text{ } = 0{,}713... + 0{,}682... ⋅ {}^{3}\!\log(x)\) Dus \(y = 0{,}71 + 0{,}68 ⋅ {}^{3}\!\log(x) \text{.}\) 1p 3p d Schrijf de formule \(y = 8 ⋅ \log(30\,000 x) - 5\) in de vorm \(y = a + b ⋅ \log(3 x) \text{.}\) Herleiden (7) 00l3 - Logaritmische formules herleiden - basis - 1ms - dynamic variables d \(y = 8 ⋅ \log(30\,000 x) - 5\) \(\text{ } = 8 ⋅ (\log(10\,000) + \log(3 x)) - 5\) 1p ○ \(\text{ } = 8 ⋅ (4 + \log(3 x)) - 5\) 1p ○ \(\text{ } = 32 + 8 ⋅ \log(3 x) - 5\) \(\text{ } = 27 + 8 ⋅ \log(3 x)\) 1p
havo wiskunde B	9.4 Formules omwerken
Logaritmische formules herleiden (6)	opgave 1 Herleid tot de gevraagde vorm. 3p a Schrijf de formule \(y = 9\,200 ⋅ 0{,}9^{x}\) 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 \(y = 9\,200 ⋅ 0{,}9^{x}\) \(\log(y) = \log(9\,200 ⋅ 0{,}9^{x})\) \(\log(y) = \log(9\,200) + \log(0{,}9^{x})\) 1p ○ \(\log(y) = \log(9\,200) + x ⋅ \log(0{,}9)\) 1p ○ \(\log(y) = 3{,}963... + x ⋅ -0{,}04575...\) Dus \(\log(y) = -0{,}0458 x + 3{,}96\) 1p 3p b Schrijf de formule \(y = 3\,100 ⋅ 1{,}25^{3 x + 2}\) 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 \(y = 3\,100 ⋅ 1{,}25^{3 x + 2}\) \(\log(y) = \log(3\,100 ⋅ 1{,}25^{3 x + 2})\) \(\log(y) = \log(3\,100) + \log(1{,}25^{3 x + 2})\) 1p ○ \(\log(y) = \log(3\,100) + (3 x + 2) ⋅ \log(1{,}25)\) \(\log(y) = \log(3\,100) + 3 x ⋅ \log(1{,}25) + 2 ⋅ \log(1{,}25)\) 1p ○ \(\log(y) = 3{,}491... + 3 x ⋅ 0{,}09691... + 2 ⋅ 0{,}09691...\) \(\log(y) = 3{,}491... + 0{,}29073... ⋅ x + 0{,}19382...\) Dus \(\log(y) = 0{,}2907 x + 3{,}69\) 1p 3p c Schrijf de formule \(\log(y) = -0{,}6526 x + 3{,}85\) 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 \(\log(y) = -0{,}6526 x + 3{,}85\) \(y = 10^{-0{,}6526 x + 3{,}85}\) 1p ○ \(y = 10^{-0{,}6526 x} ⋅ 10^{3{,}85}\) \(y = (10^{-0{,}6526})^{x} ⋅ 10^{3{,}85}\) 1p ○ \(y = 0{,}222...^{x} ⋅ 7079{,}457...\) Dus \(y = 7\,079 ⋅ 0{,}22^{x} \text{.}\) 1p 3p d Schrijf de formule \(\log(y) = 2{,}66 - 1{,}97 ⋅ \log(x)\) in de vorm \(y = a x^{b} \text{.}\) Geef \(a\) in gehelen. Dubbel (3) 00kr - Logaritmische formules herleiden - basis - 0ms - dynamic variables d \(\log(y) = 2{,}66 - 1{,}97 ⋅ \log(x)\) \(\log(y) = \log(10^{2{,}66}) + \log(x^{-1{,}97})\) \(\log(y) = \log(10^{2{,}66} ⋅ x^{-1{,}97})\) 1p ○ \(y = 10^{2{,}66} ⋅ x^{-1{,}97}\) 1p ○ \(y = 457{,}088... ⋅ x^{-1{,}97}\) Dus \(y = 457 ⋅ x^{-1{,}97} \text{.}\) 1p opgave 2 Herleid tot de gevraagde vorm. 3p a Schrijf de formule \(y = 270 x^{1{,}63}\) 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 \(y = 270 x^{1{,}63}\) \(\log(y) = \log(270 x^{1{,}63})\) 1p ○ \(\log(y) = \log(270) + \log(x^{1{,}63})\) \(\log(y) = \log(270) + 1{,}63 ⋅ \log(x)\) 1p ○ \(\log(y) = 2{,}431... + 1{,}63 ⋅ \log(x)\) Dus \(y = 2{,}43 + 1{,}63 ⋅ \log(x) \text{.}\) 1p 3p b Schrijf de formule \(y = {160 \over \sqrt{x}}\) 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 \(y = {160 \over \sqrt{x}} = 160 x^{-0{,}5}\) \(\log(y) = \log(160 x^{-0{,}5})\) 1p ○ \(\log(y) = \log(160) + \log(x^{-0{,}5})\) \(\log(y) = \log(160) - 0{,}5 ⋅ \log(x)\) 1p ○ \(\log(y) = 2{,}204... - 0{,}5 ⋅ \log(x)\) Dus \(y = 2{,}20 - 0{,}5 ⋅ \log(x) \text{.}\) 1p

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