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 \(y = 24 + 4 ⋅ {}^{7}\!\log(8 x + 5)\) Vrijmaken 00kn - Logaritmische formules herleiden - basis - 0ms - dynamic variables ○ \(y = 24 + 4 ⋅ {}^{7}\!\log(8 x + 5)\) \(4 ⋅ {}^{7}\!\log(8 x + 5) = y - 24\) \({}^{7}\!\log(8 x + 5) = \frac{1}{4} y - 6\) 1p ○ \(8 x + 5 = 7^{\frac{1}{4} y - 6}\) 1p ○ \(8 x = 7^{\frac{1}{4} y - 6} - 5\) \(x = \frac{1}{8} ⋅ 7^{\frac{1}{4} y - 6} - \frac{5}{8}\) 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 \(y = 3{,}23 ⋅ {}^{3}\!\log(x) - 2{,}32\) 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 \(y = 3{,}23 ⋅ {}^{3}\!\log(x) - 2{,}32\) \(\text{ } = {}^{3}\!\log(x^{3{,}23}) - 2{,}32\) 1p ○ \(\text{ } = {}^{3}\!\log(x^{3{,}23}) + {}^{3}\!\log(3^{-2{,}32})\) \(\text{ } = {}^{3}\!\log(x^{3{,}23} ⋅ 3^{-2{,}32})\) 1p ○ \(\text{ } = {}^{3}\!\log(x^{3{,}23} ⋅ 0{,}078...)\) Dus \(y = {}^{3}\!\log(0{,}08 ⋅ x^{3{,}23}) \text{.}\) 1p 3p b Schrijf de formule \(y = {}^{4}\!\log(28 x^{3} \sqrt{x})\) in de vorm \(y = a + b ⋅ {}^{4}\!\log(x) \text{.}\) Geef \(a\) in twee decimalen. Herleiden (5) 00l1 - Logaritmische formules herleiden - basis - 0ms - dynamic variables b \(y = {}^{4}\!\log(28 x^{3} \sqrt{x})\) \(\text{ } = {}^{4}\!\log(28 x^{3{,}5})\) 1p ○ \(\text{ } = {}^{4}\!\log(28) + {}^{4}\!\log(x^{3{,}5})\) \(\text{ } = {}^{4}\!\log(28) + 3{,}5 ⋅ {}^{4}\!\log(x)\) 1p ○ \(\text{ } = 2{,}403... + 3{,}5 ⋅ {}^{4}\!\log(x)\) Dus \(y = 2{,}40 + 3{,}5 ⋅ {}^{4}\!\log(x) \text{.}\) 1p 3p c Schrijf de formule \(y = {}^{4}\!\log(1{,}9 x) + 1{,}7\) 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 \(y = {}^{4}\!\log(1{,}9 x) + 1{,}7\) \(\text{ } = {}^{4}\!\log(1{,}9) + {}^{4}\!\log(x) + 1{,}7\) 1p ○ \(\text{ } = {}^{4}\!\log(1{,}9) + 1{,}7 + {{}^{5}\!\log(x) \over {}^{5}\!\log(4)}\) \(\text{ } = {}^{4}\!\log(1{,}9) + 1{,}7 + {1 \over {}^{5}\!\log(4)} ⋅ {}^{5}\!\log(x)\) 1p ○ \(\text{ } = 0{,}462... + 1{,}7 + {1 \over 0{,}861...} ⋅ {}^{5}\!\log(x)\) \(\text{ } = 2{,}162... + 1{,}160... ⋅ {}^{5}\!\log(x)\) Dus \(y = 2{,}16 + 1{,}16 ⋅ {}^{5}\!\log(x) \text{.}\) 1p 3p d Schrijf de formule \(y = 9 ⋅ {}^{2}\!\log(48 x) + 8\) in de vorm \(y = a + b ⋅ {}^{2}\!\log(3 x) \text{.}\) Herleiden (7) 00l3 - Logaritmische formules herleiden - basis - 0ms - dynamic variables d \(y = 9 ⋅ {}^{2}\!\log(48 x) + 8\) \(\text{ } = 9 ⋅ ({}^{2}\!\log(16) + {}^{2}\!\log(3 x)) + 8\) 1p ○ \(\text{ } = 9 ⋅ (4 + {}^{2}\!\log(3 x)) + 8\) 1p ○ \(\text{ } = 36 + 9 ⋅ {}^{2}\!\log(3 x) + 8\) \(\text{ } = 44 + 9 ⋅ {}^{2}\!\log(3 x)\) 1p

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