Logaritmische formules herleiden
Herleid tot de gevraagde vorm.
3p
Schrijf de formule \(y = 330 x^{-1{,}66}\) in de vorm \(\log(y) = a + b ⋅ \log(x) \text{.}\)
Geef \(a\) in twee decimalen.
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\(y = 330 x^{-1{,}66}\)
\(\log(y) = \log(330 x^{-1{,}66})\)
1p
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\(\log(y) = \log(330) + \log(x^{-1{,}66})\)
\(\log(y) = \log(330) - 1{,}66 ⋅ \log(x)\)
1p
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\(\log(y) = 2{,}518... - 1{,}66 ⋅ \log(x)\)
Dus \(y = 2{,}52 - 1{,}66 ⋅ \log(x) \text{.}\)
1p
Herleid tot de gevraagde vorm.
3p
Schrijf de formule \(y = {850 \over x^{2} \sqrt{x}}\) in de vorm \(\log(y) = a + b ⋅ \log(x) \text{.}\)
Geef \(a\) in twee decimalen.
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\(y = {850 \over x^{2} \sqrt{x}} = 850 x^{-2{,}5}\)
\(\log(y) = \log(850 x^{-2{,}5})\)
1p
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\(\log(y) = \log(850) + \log(x^{-2{,}5})\)
\(\log(y) = \log(850) - 2{,}5 ⋅ \log(x)\)
1p
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\(\log(y) = 2{,}929... - 2{,}5 ⋅ \log(x)\)
Dus \(y = 2{,}93 - 2{,}5 ⋅ \log(x) \text{.}\)
1p
Herleid tot de gevraagde vorm.
3p
Schrijf de formule \(\log(y) = 3{,}12 + 1{,}64 ⋅ \log(x)\) in de vorm \(y = a x^{b} \text{.}\)
Geef \(a\) in gehelen.
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\(\log(y) = 3{,}12 + 1{,}64 ⋅ \log(x)\)
\(\log(y) = \log(10^{3{,}12}) + \log(x^{1{,}64})\)
\(\log(y) = \log(10^{3{,}12} ⋅ x^{1{,}64})\)
1p
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\(y = 10^{3{,}12} ⋅ x^{1{,}64}\)
1p
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\(y = 1318{,}256... ⋅ x^{1{,}64}\)
Dus \(y = 1\,318 ⋅ x^{1{,}64} \text{.}\)
1p
Herleid tot de gevraagde vorm.
3p
Schrijf de formule \(y = 9\,100 ⋅ 1{,}27^{x}\) in de vorm \(\log(y) = a x + b \text{.}\)
Geef \(a\) in vier decimalen en \(b\) in twee decimalen.
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\(y = 9\,100 ⋅ 1{,}27^{x}\)
\(\log(y) = \log(9\,100 ⋅ 1{,}27^{x})\)
\(\log(y) = \log(9\,100) + \log(1{,}27^{x})\)
1p
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\(\log(y) = \log(9\,100) + x ⋅ \log(1{,}27)\)
1p
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\(\log(y) = 3{,}959... + x ⋅ 0{,}10380...\)
Dus \(\log(y) = 0{,}1038 x + 3{,}96\)
1p
Herleid tot de gevraagde vorm.
3p
Schrijf de formule \(y = 8\,300 ⋅ 0{,}72^{2 x + 5}\) in de vorm \(\log(y) = a x + b \text{.}\)
Geef \(a\) in vier decimalen en \(b\) in twee decimalen.
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\(y = 8\,300 ⋅ 0{,}72^{2 x + 5}\)
\(\log(y) = \log(8\,300 ⋅ 0{,}72^{2 x + 5})\)
\(\log(y) = \log(8\,300) + \log(0{,}72^{2 x + 5})\)
1p
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\(\log(y) = \log(8\,300) + (2 x + 5) ⋅ \log(0{,}72)\)
\(\log(y) = \log(8\,300) + 2 x ⋅ \log(0{,}72) + 5 ⋅ \log(0{,}72)\)
1p
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\(\log(y) = 3{,}919... + 2 x ⋅ -0{,}14266... + 5 ⋅ -0{,}14266...\)
\(\log(y) = 3{,}919... - 0{,}28533... ⋅ x - 0{,}71333...\)
Dus \(\log(y) = -0{,}2853 x + 3{,}21\)
1p
Herleid tot de gevraagde vorm.
3p
Schrijf de formule \(\log(y) = -0{,}8203 x + 3{,}39\) in de vorm \(y = b ⋅ g^{x} \text{.}\)
Geef \(b\) in gehelen en \(g\) in twee decimalen.
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\(\log(y) = -0{,}8203 x + 3{,}39\)
\(y = 10^{-0{,}8203 x + 3{,}39}\)
1p
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\(y = 10^{-0{,}8203 x} ⋅ 10^{3{,}39}\)
\(y = (10^{-0{,}8203})^{x} ⋅ 10^{3{,}39}\)
1p
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\(y = 0{,}151...^{x} ⋅ 2454{,}708...\)
Dus \(y = 2\,455 ⋅ 0{,}15^{x} \text{.}\)
1p
Herleid tot de gevraagde vorm.
3p
Schrijf de formule \(y = 1{,}37 ⋅ {}^{3}\!\log(x) - 2{,}52\) in de vorm \(y = {}^{3}\!\log(a x^{b}) \text{.}\)
Geef \(a\) en \(b\) in twee decimalen.
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\(y = 1{,}37 ⋅ {}^{3}\!\log(x) - 2{,}52\)
\(\text{ } = {}^{3}\!\log(x^{1{,}37}) - 2{,}52\)
1p
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\(\text{ } = {}^{3}\!\log(x^{1{,}37}) + {}^{3}\!\log(3^{-2{,}52})\)
\(\text{ } = {}^{3}\!\log(x^{1{,}37} ⋅ 3^{-2{,}52})\)
1p
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\(\text{ } = {}^{3}\!\log(x^{1{,}37} ⋅ 0{,}062...)\)
Dus \(y = {}^{3}\!\log(0{,}06 ⋅ x^{1{,}37}) \text{.}\)
1p
Herleid tot de gevraagde vorm.
3p
Schrijf de formule \(y = {}^{5}\!\log(81 x^{3})\) in de vorm \(y = a + b ⋅ {}^{5}\!\log(x) \text{.}\)
Geef \(a\) in twee decimalen.
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\(y = {}^{5}\!\log(81 x^{3})\)
\(\text{ } = {}^{5}\!\log(81 x^{3})\)
1p
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\(\text{ } = {}^{5}\!\log(81) + {}^{5}\!\log(x^{3})\)
\(\text{ } = {}^{5}\!\log(81) + 3 ⋅ {}^{5}\!\log(x)\)
1p
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\(\text{ } = 2{,}730... + 3 ⋅ {}^{5}\!\log(x)\)
Dus \(y = 2{,}73 + 3 ⋅ {}^{5}\!\log(x) \text{.}\)
1p
Herleid tot de gevraagde vorm.
3p
Schrijf de formule \(y = {}^{3}\!\log(2{,}2 x) - 2{,}6\) in de vorm \(y = a + b ⋅ {}^{2}\!\log(x) \text{.}\)
Geef \(a\) en \(b\) in twee decimalen.
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\(y = {}^{3}\!\log(2{,}2 x) - 2{,}6\)
\(\text{ } = {}^{3}\!\log(2{,}2) + {}^{3}\!\log(x) - 2{,}6\)
1p
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\(\text{ } = {}^{3}\!\log(2{,}2) - 2{,}6 + {{}^{2}\!\log(x) \over {}^{2}\!\log(3)}\)
\(\text{ } = {}^{3}\!\log(2{,}2) - 2{,}6 + {1 \over {}^{2}\!\log(3)} ⋅ {}^{2}\!\log(x)\)
1p
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\(\text{ } = 0{,}717... - 2{,}6 + {1 \over 1{,}584...} ⋅ {}^{2}\!\log(x)\)
\(\text{ } = -1{,}882... + 0{,}630... ⋅ {}^{2}\!\log(x)\)
Dus \(y = -1{,}88 + 0{,}63 ⋅ {}^{2}\!\log(x) \text{.}\)
1p
Herleid tot de gevraagde vorm.
3p
Schrijf de formule \(y = 6 ⋅ \log(4\,000 x) + 2\) in de vorm \(y = a + b ⋅ \log(4 x) \text{.}\)
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\(y = 6 ⋅ \log(4\,000 x) + 2\)
\(\text{ } = 6 ⋅ (\log(1\,000) + \log(4 x)) + 2\)
1p
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\(\text{ } = 6 ⋅ (3 + \log(4 x)) + 2\)
1p
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\(\text{ } = 18 + 6 ⋅ \log(4 x) + 2\)
\(\text{ } = 20 + 6 ⋅ \log(4 x)\)
1p
Druk \(x\) uit in \(y \text{.}\)
3p
\(y = 8 + 2 ⋅ {}^{3}\!\log(6 x + 8)\)
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\(y = 8 + 2 ⋅ {}^{3}\!\log(6 x + 8)\)
\(2 ⋅ {}^{3}\!\log(6 x + 8) = y - 8\)
\({}^{3}\!\log(6 x + 8) = \frac{1}{2} y - 4\)
1p
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\(6 x + 8 = 3^{\frac{1}{2} y - 4}\)
1p
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\(6 x = 3^{\frac{1}{2} y - 4} - 8\)
\(x = \frac{1}{6} ⋅ 3^{\frac{1}{2} y - 4} - 1\frac{1}{3}\)
1p