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 \(y = 15 + 3 ⋅ {}^{7}\!\log(9 x + 8)\) Vrijmaken 00kn - Logaritmische formules herleiden - basis - 0ms - dynamic variables ○ \(y = 15 + 3 ⋅ {}^{7}\!\log(9 x + 8)\) \(3 ⋅ {}^{7}\!\log(9 x + 8) = y - 15\) \({}^{7}\!\log(9 x + 8) = \frac{1}{3} y - 5\) 1p ○ \(9 x + 8 = 7^{\frac{1}{3} y - 5}\) 1p ○ \(9 x = 7^{\frac{1}{3} y - 5} - 8\) \(x = \frac{1}{9} ⋅ 7^{\frac{1}{3} y - 5} - \frac{8}{9}\) 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 \(y = 1{,}12 ⋅ {}^{5}\!\log(x) + 1{,}27\) 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 \(y = 1{,}12 ⋅ {}^{5}\!\log(x) + 1{,}27\) \(\text{ } = {}^{5}\!\log(x^{1{,}12}) + 1{,}27\) 1p ○ \(\text{ } = {}^{5}\!\log(x^{1{,}12}) + {}^{5}\!\log(5^{1{,}27})\) \(\text{ } = {}^{5}\!\log(x^{1{,}12} ⋅ 5^{1{,}27})\) 1p ○ \(\text{ } = {}^{5}\!\log(x^{1{,}12} ⋅ 7{,}721...)\) Dus \(y = {}^{5}\!\log(7{,}72 ⋅ x^{1{,}12}) \text{.}\) 1p 3p b Schrijf de formule \(y = {}^{2}\!\log(51 x^{4})\) 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 \(y = {}^{2}\!\log(51 x^{4})\) \(\text{ } = {}^{2}\!\log(51 x^{4})\) 1p ○ \(\text{ } = {}^{2}\!\log(51) + {}^{2}\!\log(x^{4})\) \(\text{ } = {}^{2}\!\log(51) + 4 ⋅ {}^{2}\!\log(x)\) 1p ○ \(\text{ } = 5{,}672... + 4 ⋅ {}^{2}\!\log(x)\) Dus \(y = 5{,}67 + 4 ⋅ {}^{2}\!\log(x) \text{.}\) 1p 3p c Schrijf de formule \(y = {}^{4}\!\log(1{,}9 x) + 0{,}3\) 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 \(y = {}^{4}\!\log(1{,}9 x) + 0{,}3\) \(\text{ } = {}^{4}\!\log(1{,}9) + {}^{4}\!\log(x) + 0{,}3\) 1p ○ \(\text{ } = {}^{4}\!\log(1{,}9) + 0{,}3 + {{}^{5}\!\log(x) \over {}^{5}\!\log(4)}\) \(\text{ } = {}^{4}\!\log(1{,}9) + 0{,}3 + {1 \over {}^{5}\!\log(4)} ⋅ {}^{5}\!\log(x)\) 1p ○ \(\text{ } = 0{,}462... + 0{,}3 + {1 \over 0{,}861...} ⋅ {}^{5}\!\log(x)\) \(\text{ } = 0{,}762... + 1{,}160... ⋅ {}^{5}\!\log(x)\) Dus \(y = 0{,}76 + 1{,}16 ⋅ {}^{5}\!\log(x) \text{.}\) 1p 3p d Schrijf de formule \(y = 6 ⋅ {}^{3}\!\log(36 x) + 10\) in de vorm \(y = a + b ⋅ {}^{3}\!\log(4 x) \text{.}\) Herleiden (7) 00l3 - Logaritmische formules herleiden - basis - 1ms - dynamic variables d \(y = 6 ⋅ {}^{3}\!\log(36 x) + 10\) \(\text{ } = 6 ⋅ ({}^{3}\!\log(9) + {}^{3}\!\log(4 x)) + 10\) 1p ○ \(\text{ } = 6 ⋅ (2 + {}^{3}\!\log(4 x)) + 10\) 1p ○ \(\text{ } = 12 + 6 ⋅ {}^{3}\!\log(4 x) + 10\) \(\text{ } = 22 + 6 ⋅ {}^{3}\!\log(4 x)\) 1p

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