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 \(y = 20 + 4 ⋅ {}^{7}\!\log(8 x - 9)\) Vrijmaken 00kn - Logaritmische formules herleiden - basis - 0ms - dynamic variables ○ \(y = 20 + 4 ⋅ {}^{7}\!\log(8 x - 9)\) \(4 ⋅ {}^{7}\!\log(8 x - 9) = y - 20\) \({}^{7}\!\log(8 x - 9) = \frac{1}{4} y - 5\) 1p ○ \(8 x - 9 = 7^{\frac{1}{4} y - 5}\) 1p ○ \(8 x = 7^{\frac{1}{4} y - 5} + 9\) \(x = \frac{1}{8} ⋅ 7^{\frac{1}{4} y - 5} + 1\frac{1}{8}\) 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 \(y = 3{,}52 ⋅ {}^{3}\!\log(x) + 1{,}76\) in de vorm \(y = {}^{3}\!\log(a x^{b}) \text{.}\) Geef \(a\) en \(b\) in twee decimalen. Herleiden (4) 00l0 - Logaritmische formules herleiden - basis - 0ms - dynamic variables a \(y = 3{,}52 ⋅ {}^{3}\!\log(x) + 1{,}76\) \(\text{ } = {}^{3}\!\log(x^{3{,}52}) + 1{,}76\) 1p ○ \(\text{ } = {}^{3}\!\log(x^{3{,}52}) + {}^{3}\!\log(3^{1{,}76})\) \(\text{ } = {}^{3}\!\log(x^{3{,}52} ⋅ 3^{1{,}76})\) 1p ○ \(\text{ } = {}^{3}\!\log(x^{3{,}52} ⋅ 6{,}914...)\) Dus \(y = {}^{3}\!\log(6{,}91 ⋅ x^{3{,}52}) \text{.}\) 1p 3p b Schrijf de formule \(y = {}^{5}\!\log({59 \over x^{3}})\) 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 \(y = {}^{5}\!\log({59 \over x^{3}})\) \(\text{ } = {}^{5}\!\log(59 x^{-3})\) 1p ○ \(\text{ } = {}^{5}\!\log(59) + {}^{5}\!\log(x^{-3})\) \(\text{ } = {}^{5}\!\log(59) - 3 ⋅ {}^{5}\!\log(x)\) 1p ○ \(\text{ } = 2{,}533... - 3 ⋅ {}^{5}\!\log(x)\) Dus \(y = 2{,}53 - 3 ⋅ {}^{5}\!\log(x) \text{.}\) 1p 3p c Schrijf de formule \(y = {}^{4}\!\log(1{,}7 x) + 1{,}2\) 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 \(y = {}^{4}\!\log(1{,}7 x) + 1{,}2\) \(\text{ } = {}^{4}\!\log(1{,}7) + {}^{4}\!\log(x) + 1{,}2\) 1p ○ \(\text{ } = {}^{4}\!\log(1{,}7) + 1{,}2 + {{}^{3}\!\log(x) \over {}^{3}\!\log(4)}\) \(\text{ } = {}^{4}\!\log(1{,}7) + 1{,}2 + {1 \over {}^{3}\!\log(4)} ⋅ {}^{3}\!\log(x)\) 1p ○ \(\text{ } = 0{,}382... + 1{,}2 + {1 \over 1{,}261...} ⋅ {}^{3}\!\log(x)\) \(\text{ } = 1{,}582... + 0{,}792... ⋅ {}^{3}\!\log(x)\) Dus \(y = 1{,}58 + 0{,}79 ⋅ {}^{3}\!\log(x) \text{.}\) 1p 3p d Schrijf de formule \(y = 7 ⋅ \log(30\,000 x) + 5\) in de vorm \(y = a + b ⋅ \log(3 x) \text{.}\) Herleiden (7) 00l3 - Logaritmische formules herleiden - basis - 0ms - dynamic variables d \(y = 7 ⋅ \log(30\,000 x) + 5\) \(\text{ } = 7 ⋅ (\log(10\,000) + \log(3 x)) + 5\) 1p ○ \(\text{ } = 7 ⋅ (4 + \log(3 x)) + 5\) 1p ○ \(\text{ } = 28 + 7 ⋅ \log(3 x) + 5\) \(\text{ } = 33 + 7 ⋅ \log(3 x)\) 1p
havo wiskunde B	9.4 Formules omwerken
Logaritmische formules herleiden (6)	opgave 1 Herleid tot de gevraagde vorm. 3p a Schrijf de formule \(y = 7\,300 ⋅ 0{,}92^{x}\) 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 \(y = 7\,300 ⋅ 0{,}92^{x}\) \(\log(y) = \log(7\,300 ⋅ 0{,}92^{x})\) \(\log(y) = \log(7\,300) + \log(0{,}92^{x})\) 1p ○ \(\log(y) = \log(7\,300) + x ⋅ \log(0{,}92)\) 1p ○ \(\log(y) = 3{,}863... + x ⋅ -0{,}03621...\) Dus \(\log(y) = -0{,}0362 x + 3{,}86\) 1p 3p b Schrijf de formule \(y = 5\,900 ⋅ 0{,}94^{3 x + 2}\) 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 \(y = 5\,900 ⋅ 0{,}94^{3 x + 2}\) \(\log(y) = \log(5\,900 ⋅ 0{,}94^{3 x + 2})\) \(\log(y) = \log(5\,900) + \log(0{,}94^{3 x + 2})\) 1p ○ \(\log(y) = \log(5\,900) + (3 x + 2) ⋅ \log(0{,}94)\) \(\log(y) = \log(5\,900) + 3 x ⋅ \log(0{,}94) + 2 ⋅ \log(0{,}94)\) 1p ○ \(\log(y) = 3{,}770... + 3 x ⋅ -0{,}02687... + 2 ⋅ -0{,}02687...\) \(\log(y) = 3{,}770... - 0{,}08061... ⋅ x - 0{,}05374...\) Dus \(\log(y) = -0{,}0806 x + 3{,}72\) 1p 3p c Schrijf de formule \(\log(y) = 0{,}1205 x + 3{,}79\) 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 \(\log(y) = 0{,}1205 x + 3{,}79\) \(y = 10^{0{,}1205 x + 3{,}79}\) 1p ○ \(y = 10^{0{,}1205 x} ⋅ 10^{3{,}79}\) \(y = (10^{0{,}1205})^{x} ⋅ 10^{3{,}79}\) 1p ○ \(y = 1{,}319...^{x} ⋅ 6165{,}950...\) Dus \(y = 6\,166 ⋅ 1{,}32^{x} \text{.}\) 1p 3p d Schrijf de formule \(\log(y) = 1{,}12 + 1{,}16 ⋅ \log(x)\) in de vorm \(y = a x^{b} \text{.}\) Geef \(a\) in gehelen. Dubbel (3) 00kr - Logaritmische formules herleiden - basis - 0ms - dynamic variables d \(\log(y) = 1{,}12 + 1{,}16 ⋅ \log(x)\) \(\log(y) = \log(10^{1{,}12}) + \log(x^{1{,}16})\) \(\log(y) = \log(10^{1{,}12} ⋅ x^{1{,}16})\) 1p ○ \(y = 10^{1{,}12} ⋅ x^{1{,}16}\) 1p ○ \(y = 13{,}182... ⋅ x^{1{,}16}\) Dus \(y = 13 ⋅ x^{1{,}16} \text{.}\) 1p opgave 2 Herleid tot de gevraagde vorm. 3p a Schrijf de formule \(y = 700 x^{1{,}34}\) 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 \(y = 700 x^{1{,}34}\) \(\log(y) = \log(700 x^{1{,}34})\) 1p ○ \(\log(y) = \log(700) + \log(x^{1{,}34})\) \(\log(y) = \log(700) + 1{,}34 ⋅ \log(x)\) 1p ○ \(\log(y) = 2{,}845... + 1{,}34 ⋅ \log(x)\) Dus \(y = 2{,}85 + 1{,}34 ⋅ \log(x) \text{.}\) 1p 3p b Schrijf de formule \(y = {180 \over x^{5}}\) 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 \(y = {180 \over x^{5}} = 180 x^{-5}\) \(\log(y) = \log(180 x^{-5})\) 1p ○ \(\log(y) = \log(180) + \log(x^{-5})\) \(\log(y) = \log(180) - 5 ⋅ \log(x)\) 1p ○ \(\log(y) = 2{,}255... - 5 ⋅ \log(x)\) Dus \(y = 2{,}26 - 5 ⋅ \log(x) \text{.}\) 1p

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