Getal & Ruimte (13e 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 \(y = 36 + 4 ⋅ {}^{7}\!\log(3 x + 1)\) Vrijmaken 00kn - Logaritmische formules herleiden - basis - 0ms - dynamic variables ○ \(y = 36 + 4 ⋅ {}^{7}\!\log(3 x + 1)\) \(4 ⋅ {}^{7}\!\log(3 x + 1) = y - 36\) \({}^{7}\!\log(3 x + 1) = \frac{1}{4} y - 9\) 1p ○ \(3 x + 1 = 7^{\frac{1}{4} y - 9}\) 1p ○ \(3 x = 7^{\frac{1}{4} y - 9} - 1\) \(x = \frac{1}{3} ⋅ 7^{\frac{1}{4} y - 9} - \frac{1}{3}\) 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 \(y = 3{,}61 ⋅ {}^{2}\!\log(x) - 2{,}37\) 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 \(y = 3{,}61 ⋅ {}^{2}\!\log(x) - 2{,}37\) \(\text{ } = {}^{2}\!\log(x^{3{,}61}) - 2{,}37\) 1p ○ \(\text{ } = {}^{2}\!\log(x^{3{,}61}) + {}^{2}\!\log(2^{-2{,}37})\) \(\text{ } = {}^{2}\!\log(x^{3{,}61} ⋅ 2^{-2{,}37})\) 1p ○ \(\text{ } = {}^{2}\!\log(x^{3{,}61} ⋅ 0{,}193...)\) Dus \(y = {}^{2}\!\log(0{,}19 ⋅ x^{3{,}61}) \text{.}\) 1p 3p b Schrijf de formule \(y = {}^{2}\!\log(100 x^{5})\) in de vorm \(y = a + b ⋅ {}^{2}\!\log(x) \text{.}\) Geef \(a\) in twee decimalen. Herleiden (5) 00l1 - Logaritmische formules herleiden - basis - 0ms - dynamic variables b \(y = {}^{2}\!\log(100 x^{5})\) \(\text{ } = {}^{2}\!\log(100 x^{5})\) 1p ○ \(\text{ } = {}^{2}\!\log(100) + {}^{2}\!\log(x^{5})\) \(\text{ } = {}^{2}\!\log(100) + 5 ⋅ {}^{2}\!\log(x)\) 1p ○ \(\text{ } = 6{,}643... + 5 ⋅ {}^{2}\!\log(x)\) Dus \(y = 6{,}64 + 5 ⋅ {}^{2}\!\log(x) \text{.}\) 1p 3p c Schrijf de formule \(y = {}^{2}\!\log(2{,}6 x) - 1{,}5\) 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 \(y = {}^{2}\!\log(2{,}6 x) - 1{,}5\) \(\text{ } = {}^{2}\!\log(2{,}6) + {}^{2}\!\log(x) - 1{,}5\) 1p ○ \(\text{ } = {}^{2}\!\log(2{,}6) - 1{,}5 + {{}^{5}\!\log(x) \over {}^{5}\!\log(2)}\) \(\text{ } = {}^{2}\!\log(2{,}6) - 1{,}5 + {1 \over {}^{5}\!\log(2)} ⋅ {}^{5}\!\log(x)\) 1p ○ \(\text{ } = 1{,}378... - 1{,}5 + {1 \over 0{,}430...} ⋅ {}^{5}\!\log(x)\) \(\text{ } = -0{,}121... + 2{,}321... ⋅ {}^{5}\!\log(x)\) Dus \(y = -0{,}12 + 2{,}32 ⋅ {}^{5}\!\log(x) \text{.}\) 1p 3p d Schrijf de formule \(y = 6 ⋅ {}^{2}\!\log(32 x) + 10\) in de vorm \(y = a + b ⋅ {}^{2}\!\log(4 x) \text{.}\) Herleiden (7) 00l3 - Logaritmische formules herleiden - basis - 1ms - dynamic variables d \(y = 6 ⋅ {}^{2}\!\log(32 x) + 10\) \(\text{ } = 6 ⋅ ({}^{2}\!\log(8) + {}^{2}\!\log(4 x)) + 10\) 1p ○ \(\text{ } = 6 ⋅ (3 + {}^{2}\!\log(4 x)) + 10\) 1p ○ \(\text{ } = 18 + 6 ⋅ {}^{2}\!\log(4 x) + 10\) \(\text{ } = 28 + 6 ⋅ {}^{2}\!\log(4 x)\) 1p

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