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 \(y = 18 + 3 ⋅ {}^{9}\!\log(7 x + 2)\) Vrijmaken 00kn - Logaritmische formules herleiden - basis - 0ms - dynamic variables ○ \(y = 18 + 3 ⋅ {}^{9}\!\log(7 x + 2)\) \(3 ⋅ {}^{9}\!\log(7 x + 2) = y - 18\) \({}^{9}\!\log(7 x + 2) = \frac{1}{3} y - 6\) 1p ○ \(7 x + 2 = 9^{\frac{1}{3} y - 6}\) 1p ○ \(7 x = 9^{\frac{1}{3} y - 6} - 2\) \(x = \frac{1}{7} ⋅ 9^{\frac{1}{3} y - 6} - \frac{2}{7}\) 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 \(y = 2{,}78 ⋅ {}^{5}\!\log(x) - 1{,}46\) in de vorm \(y = {}^{5}\!\log(a x^{b}) \text{.}\) Geef \(a\) en \(b\) in twee decimalen. Herleiden (4) 00l0 - Logaritmische formules herleiden - basis - 0ms - dynamic variables a \(y = 2{,}78 ⋅ {}^{5}\!\log(x) - 1{,}46\) \(\text{ } = {}^{5}\!\log(x^{2{,}78}) - 1{,}46\) 1p ○ \(\text{ } = {}^{5}\!\log(x^{2{,}78}) + {}^{5}\!\log(5^{-1{,}46})\) \(\text{ } = {}^{5}\!\log(x^{2{,}78} ⋅ 5^{-1{,}46})\) 1p ○ \(\text{ } = {}^{5}\!\log(x^{2{,}78} ⋅ 0{,}095...)\) Dus \(y = {}^{5}\!\log(0{,}10 ⋅ x^{2{,}78}) \text{.}\) 1p 3p b Schrijf de formule \(y = {}^{5}\!\log(96 x^{4} \sqrt{x})\) 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 \(y = {}^{5}\!\log(96 x^{4} \sqrt{x})\) \(\text{ } = {}^{5}\!\log(96 x^{4{,}5})\) 1p ○ \(\text{ } = {}^{5}\!\log(96) + {}^{5}\!\log(x^{4{,}5})\) \(\text{ } = {}^{5}\!\log(96) + 4{,}5 ⋅ {}^{5}\!\log(x)\) 1p ○ \(\text{ } = 2{,}835... + 4{,}5 ⋅ {}^{5}\!\log(x)\) Dus \(y = 2{,}84 + 4{,}5 ⋅ {}^{5}\!\log(x) \text{.}\) 1p 3p c Schrijf de formule \(y = {}^{4}\!\log(1{,}7 x) - 2{,}2\) in de vorm \(y = a + b ⋅ {}^{5}\!\log(x) \text{.}\) Geef \(a\) en \(b\) in twee decimalen. Herleiden (6) 00l2 - Logaritmische formules herleiden - basis - 0ms - dynamic variables c \(y = {}^{4}\!\log(1{,}7 x) - 2{,}2\) \(\text{ } = {}^{4}\!\log(1{,}7) + {}^{4}\!\log(x) - 2{,}2\) 1p ○ \(\text{ } = {}^{4}\!\log(1{,}7) - 2{,}2 + {{}^{5}\!\log(x) \over {}^{5}\!\log(4)}\) \(\text{ } = {}^{4}\!\log(1{,}7) - 2{,}2 + {1 \over {}^{5}\!\log(4)} ⋅ {}^{5}\!\log(x)\) 1p ○ \(\text{ } = 0{,}382... - 2{,}2 + {1 \over 0{,}861...} ⋅ {}^{5}\!\log(x)\) \(\text{ } = -1{,}817... + 1{,}160... ⋅ {}^{5}\!\log(x)\) Dus \(y = -1{,}82 + 1{,}16 ⋅ {}^{5}\!\log(x) \text{.}\) 1p 3p d Schrijf de formule \(y = 7 ⋅ {}^{4}\!\log(128 x) - 8\) in de vorm \(y = a + b ⋅ {}^{4}\!\log(2 x) \text{.}\) Herleiden (7) 00l3 - Logaritmische formules herleiden - basis - 1ms - dynamic variables d \(y = 7 ⋅ {}^{4}\!\log(128 x) - 8\) \(\text{ } = 7 ⋅ ({}^{4}\!\log(64) + {}^{4}\!\log(2 x)) - 8\) 1p ○ \(\text{ } = 7 ⋅ (3 + {}^{4}\!\log(2 x)) - 8\) 1p ○ \(\text{ } = 21 + 7 ⋅ {}^{4}\!\log(2 x) - 8\) \(\text{ } = 13 + 7 ⋅ {}^{4}\!\log(2 x)\) 1p
havo wiskunde B	9.4 Formules omwerken
Logaritmische formules herleiden (6)	opgave 1 Herleid tot de gevraagde vorm. 3p a Schrijf de formule \(y = 9\,500 ⋅ 0{,}77^{x}\) 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 \(y = 9\,500 ⋅ 0{,}77^{x}\) \(\log(y) = \log(9\,500 ⋅ 0{,}77^{x})\) \(\log(y) = \log(9\,500) + \log(0{,}77^{x})\) 1p ○ \(\log(y) = \log(9\,500) + x ⋅ \log(0{,}77)\) 1p ○ \(\log(y) = 3{,}977... + x ⋅ -0{,}11350...\) Dus \(\log(y) = -0{,}1135 x + 3{,}98\) 1p 3p b Schrijf de formule \(y = 1\,000 ⋅ 1{,}13^{5 x + 6}\) 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 \(y = 1\,000 ⋅ 1{,}13^{5 x + 6}\) \(\log(y) = \log(1\,000 ⋅ 1{,}13^{5 x + 6})\) \(\log(y) = \log(1\,000) + \log(1{,}13^{5 x + 6})\) 1p ○ \(\log(y) = \log(1\,000) + (5 x + 6) ⋅ \log(1{,}13)\) \(\log(y) = \log(1\,000) + 5 x ⋅ \log(1{,}13) + 6 ⋅ \log(1{,}13)\) 1p ○ \(\log(y) = 3 + 5 x ⋅ 0{,}05307... + 6 ⋅ 0{,}05307...\) \(\log(y) = 3 + 0{,}26539... ⋅ x + 0{,}31847...\) Dus \(\log(y) = 0{,}2654 x + 3{,}32\) 1p 3p c Schrijf de formule \(\log(y) = -0{,}7482 x + 2{,}12\) 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 \(\log(y) = -0{,}7482 x + 2{,}12\) \(y = 10^{-0{,}7482 x + 2{,}12}\) 1p ○ \(y = 10^{-0{,}7482 x} ⋅ 10^{2{,}12}\) \(y = (10^{-0{,}7482})^{x} ⋅ 10^{2{,}12}\) 1p ○ \(y = 0{,}178...^{x} ⋅ 131{,}825...\) Dus \(y = 132 ⋅ 0{,}18^{x} \text{.}\) 1p 3p d Schrijf de formule \(\log(y) = 2{,}62 - 1{,}42 ⋅ \log(x)\) in de vorm \(y = a x^{b} \text{.}\) Geef \(a\) in gehelen. Dubbel (3) 00kr - Logaritmische formules herleiden - basis - 0ms - dynamic variables d \(\log(y) = 2{,}62 - 1{,}42 ⋅ \log(x)\) \(\log(y) = \log(10^{2{,}62}) + \log(x^{-1{,}42})\) \(\log(y) = \log(10^{2{,}62} ⋅ x^{-1{,}42})\) 1p ○ \(y = 10^{2{,}62} ⋅ x^{-1{,}42}\) 1p ○ \(y = 416{,}869... ⋅ x^{-1{,}42}\) Dus \(y = 417 ⋅ x^{-1{,}42} \text{.}\) 1p opgave 2 Herleid tot de gevraagde vorm. 3p a Schrijf de formule \(y = 480 x^{1{,}93}\) 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 \(y = 480 x^{1{,}93}\) \(\log(y) = \log(480 x^{1{,}93})\) 1p ○ \(\log(y) = \log(480) + \log(x^{1{,}93})\) \(\log(y) = \log(480) + 1{,}93 ⋅ \log(x)\) 1p ○ \(\log(y) = 2{,}681... + 1{,}93 ⋅ \log(x)\) Dus \(y = 2{,}68 + 1{,}93 ⋅ \log(x) \text{.}\) 1p 3p b Schrijf de formule \(y = {860 \over x^{2}}\) 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 \(y = {860 \over x^{2}} = 860 x^{-2}\) \(\log(y) = \log(860 x^{-2})\) 1p ○ \(\log(y) = \log(860) + \log(x^{-2})\) \(\log(y) = \log(860) - 2 ⋅ \log(x)\) 1p ○ \(\log(y) = 2{,}934... - 2 ⋅ \log(x)\) Dus \(y = 2{,}93 - 2 ⋅ \log(x) \text{.}\) 1p

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