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 \(y = 10 + 2 ⋅ {}^{9}\!\log(3 x + 7)\) Vrijmaken 00kn - Logaritmische formules herleiden - basis - 1ms - dynamic variables ○ \(y = 10 + 2 ⋅ {}^{9}\!\log(3 x + 7)\) \(2 ⋅ {}^{9}\!\log(3 x + 7) = y - 10\) \({}^{9}\!\log(3 x + 7) = \frac{1}{2} y - 5\) 1p ○ \(3 x + 7 = 9^{\frac{1}{2} y - 5}\) 1p ○ \(3 x = 9^{\frac{1}{2} y - 5} - 7\) \(x = \frac{1}{3} ⋅ 9^{\frac{1}{2} y - 5} - 2\frac{1}{3}\) 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 \(y = 2{,}29 ⋅ {}^{4}\!\log(x) - 2{,}55\) 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 \(y = 2{,}29 ⋅ {}^{4}\!\log(x) - 2{,}55\) \(\text{ } = {}^{4}\!\log(x^{2{,}29}) - 2{,}55\) 1p ○ \(\text{ } = {}^{4}\!\log(x^{2{,}29}) + {}^{4}\!\log(4^{-2{,}55})\) \(\text{ } = {}^{4}\!\log(x^{2{,}29} ⋅ 4^{-2{,}55})\) 1p ○ \(\text{ } = {}^{4}\!\log(x^{2{,}29} ⋅ 0{,}029...)\) Dus \(y = {}^{4}\!\log(0{,}03 ⋅ x^{2{,}29}) \text{.}\) 1p 3p b Schrijf de formule \(y = {}^{5}\!\log({48 \over x^{2} \sqrt{x}})\) in de vorm \(y = a + b ⋅ {}^{5}\!\log(x) \text{.}\) Geef \(a\) in twee decimalen. Logaritmisch (5) 00l1 - Logaritmische formules herleiden - basis - 0ms - dynamic variables b \(y = {}^{5}\!\log({48 \over x^{2} \sqrt{x}})\) \(\text{ } = {}^{5}\!\log(48 x^{-2{,}5})\) 1p ○ \(\text{ } = {}^{5}\!\log(48) + {}^{5}\!\log(x^{-2{,}5})\) \(\text{ } = {}^{5}\!\log(48) - 2{,}5 ⋅ {}^{5}\!\log(x)\) 1p ○ \(\text{ } = 2{,}405... - 2{,}5 ⋅ {}^{5}\!\log(x)\) Dus \(y = 2{,}41 - 2{,}5 ⋅ {}^{5}\!\log(x) \text{.}\) 1p 3p c Schrijf de formule \(y = {}^{2}\!\log(1{,}2 x) - 0{,}7\) 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 \(y = {}^{2}\!\log(1{,}2 x) - 0{,}7\) \(\text{ } = {}^{2}\!\log(1{,}2) + {}^{2}\!\log(x) - 0{,}7\) 1p ○ \(\text{ } = {}^{2}\!\log(1{,}2) - 0{,}7 + {{}^{3}\!\log(x) \over {}^{3}\!\log(2)}\) \(\text{ } = {}^{2}\!\log(1{,}2) - 0{,}7 + {1 \over {}^{3}\!\log(2)} ⋅ {}^{3}\!\log(x)\) 1p ○ \(\text{ } = 0{,}263... - 0{,}7 + {1 \over 0{,}630...} ⋅ {}^{3}\!\log(x)\) \(\text{ } = -0{,}436... + 1{,}584... ⋅ {}^{3}\!\log(x)\) Dus \(y = -0{,}44 + 1{,}58 ⋅ {}^{3}\!\log(x) \text{.}\) 1p 3p d Schrijf de formule \(y = 6 ⋅ \log(300 x) - 8\) in de vorm \(y = a + b ⋅ \log(3 x) \text{.}\) Herleiden (7) 00l3 - Logaritmische formules herleiden - basis - 1ms - dynamic variables d \(y = 6 ⋅ \log(300 x) - 8\) \(\text{ } = 6 ⋅ (\log(100) + \log(3 x)) - 8\) 1p ○ \(\text{ } = 6 ⋅ (2 + \log(3 x)) - 8\) 1p ○ \(\text{ } = 12 + 6 ⋅ \log(3 x) - 8\) \(\text{ } = 4 + 6 ⋅ \log(3 x)\) 1p
havo wiskunde B	9.4 Formules omwerken
Logaritmische formules herleiden (6)	opgave 1 Herleid tot de gevraagde vorm. 3p a Schrijf de formule \(y = 2\,900 ⋅ 1{,}05^{x}\) 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 \(y = 2\,900 ⋅ 1{,}05^{x}\) \(\log(y) = \log(2\,900 ⋅ 1{,}05^{x})\) \(\log(y) = \log(2\,900) + \log(1{,}05^{x})\) 1p ○ \(\log(y) = \log(2\,900) + x ⋅ \log(1{,}05)\) 1p ○ \(\log(y) = 3{,}462... + x ⋅ 0{,}02118...\) Dus \(\log(y) = 0{,}0212 x + 3{,}46\) 1p 3p b Schrijf de formule \(y = 1\,600 ⋅ 0{,}88^{2 x + 1}\) 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 - 1ms - dynamic variables b \(y = 1\,600 ⋅ 0{,}88^{2 x + 1}\) \(\log(y) = \log(1\,600 ⋅ 0{,}88^{2 x + 1})\) \(\log(y) = \log(1\,600) + \log(0{,}88^{2 x + 1})\) 1p ○ \(\log(y) = \log(1\,600) + (2 x + 1) ⋅ \log(0{,}88)\) \(\log(y) = \log(1\,600) + 2 x ⋅ \log(0{,}88) + 1 ⋅ \log(0{,}88)\) 1p ○ \(\log(y) = 3{,}204... + 2 x ⋅ -0{,}05551... + 1 ⋅ -0{,}05551...\) \(\log(y) = 3{,}204... - 0{,}11103... ⋅ x - 0{,}05551...\) Dus \(\log(y) = -0{,}1110 x + 3{,}15\) 1p 3p c Schrijf de formule \(\log(y) = 0{,}0968 x + 3{,}66\) 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 \(\log(y) = 0{,}0968 x + 3{,}66\) \(y = 10^{0{,}0968 x + 3{,}66}\) 1p ○ \(y = 10^{0{,}0968 x} ⋅ 10^{3{,}66}\) \(y = (10^{0{,}0968})^{x} ⋅ 10^{3{,}66}\) 1p ○ \(y = 1{,}249...^{x} ⋅ 4570{,}881...\) Dus \(y = 4\,571 ⋅ 1{,}25^{x} \text{.}\) 1p 3p d Schrijf de formule \(\log(y) = 2{,}95 + 1{,}44 ⋅ \log(x)\) in de vorm \(y = a x^{b} \text{.}\) Geef \(a\) in gehelen. Dubbel (3) 00kr - Logaritmische formules herleiden - basis - 0ms - dynamic variables d \(\log(y) = 2{,}95 + 1{,}44 ⋅ \log(x)\) \(\log(y) = \log(10^{2{,}95}) + \log(x^{1{,}44})\) \(\log(y) = \log(10^{2{,}95} ⋅ x^{1{,}44})\) 1p ○ \(y = 10^{2{,}95} ⋅ x^{1{,}44}\) 1p ○ \(y = 891{,}250... ⋅ x^{1{,}44}\) Dus \(y = 891 ⋅ x^{1{,}44} \text{.}\) 1p opgave 2 Herleid tot de gevraagde vorm. 3p a Schrijf de formule \(y = 540 x^{1{,}57}\) 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 \(y = 540 x^{1{,}57}\) \(\log(y) = \log(540 x^{1{,}57})\) 1p ○ \(\log(y) = \log(540) + \log(x^{1{,}57})\) \(\log(y) = \log(540) + 1{,}57 ⋅ \log(x)\) 1p ○ \(\log(y) = 2{,}732... + 1{,}57 ⋅ \log(x)\) Dus \(y = 2{,}73 + 1{,}57 ⋅ \log(x) \text{.}\) 1p 3p b Schrijf de formule \(y = {750 \over x \sqrt{x}}\) 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 \(y = {750 \over x \sqrt{x}} = 750 x^{-1{,}5}\) \(\log(y) = \log(750 x^{-1{,}5})\) 1p ○ \(\log(y) = \log(750) + \log(x^{-1{,}5})\) \(\log(y) = \log(750) - 1{,}5 ⋅ \log(x)\) 1p ○ \(\log(y) = 2{,}875... - 1{,}5 ⋅ \log(x)\) Dus \(y = 2{,}88 - 1{,}5 ⋅ \log(x) \text{.}\) 1p

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