Getal & Ruimte (12e editie) - vwo wiskunde A

'Logaritmische formules herleiden'.

vwo wiskunde A 13.4 Omvormen van formules met exponenten en logaritmen

Logaritmische formules herleiden (5)

opgave 1

Druk \(x\) uit in \(y \text{.}\)

3p

\(y = 8 + 4 ⋅ {}^{9}\!\log(7 x + 8)\)

Vrijmaken
00kn - Logaritmische formules herleiden - basis - 1ms - dynamic variables

○

\(y = 8 + 4 ⋅ {}^{9}\!\log(7 x + 8)\)
\(4 ⋅ {}^{9}\!\log(7 x + 8) = y - 8\)
\({}^{9}\!\log(7 x + 8) = \frac{1}{4} y - 2\)

1p

○

\(7 x + 8 = 9^{\frac{1}{4} y - 2}\)

1p

○

\(7 x = 9^{\frac{1}{4} y - 2} - 8\)
\(x = \frac{1}{7} ⋅ 9^{\frac{1}{4} y - 2} - 1\frac{1}{7}\)

1p

opgave 2

Herleid tot de gevraagde vorm.

3p

a

Schrijf de formule \(y = 8\,500 ⋅ 1{,}26^{x}\) 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

\(y = 8\,500 ⋅ 1{,}26^{x}\)
\(\log(y) = \log(8\,500 ⋅ 1{,}26^{x})\)
\(\log(y) = \log(8\,500) + \log(1{,}26^{x})\)

1p

○

\(\log(y) = \log(8\,500) + x ⋅ \log(1{,}26)\)

1p

○

\(\log(y) = 3{,}929... + x ⋅ 0{,}10037...\)
Dus \(\log(y) = 0{,}1004 x + 3{,}93\)

1p

3p

b

Schrijf de formule \(y = 4\,000 ⋅ 1{,}05^{4 x + 5}\) 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 - 1ms - dynamic variables

b

\(y = 4\,000 ⋅ 1{,}05^{4 x + 5}\)
\(\log(y) = \log(4\,000 ⋅ 1{,}05^{4 x + 5})\)
\(\log(y) = \log(4\,000) + \log(1{,}05^{4 x + 5})\)

1p

○

\(\log(y) = \log(4\,000) + (4 x + 5) ⋅ \log(1{,}05)\)
\(\log(y) = \log(4\,000) + 4 x ⋅ \log(1{,}05) + 5 ⋅ \log(1{,}05)\)

1p

○

\(\log(y) = 3{,}602... + 4 x ⋅ 0{,}02118... + 5 ⋅ 0{,}02118...\)
\(\log(y) = 3{,}602... + 0{,}08475... ⋅ x + 0{,}10594...\)
Dus \(\log(y) = 0{,}0848 x + 3{,}71\)

1p

3p

c

Schrijf de formule \(\log(y) = 0{,}2919 x + 3{,}96\) 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

\(\log(y) = 0{,}2919 x + 3{,}96\)
\(y = 10^{0{,}2919 x + 3{,}96}\)

1p

○

\(y = 10^{0{,}2919 x} ⋅ 10^{3{,}96}\)
\(y = (10^{0{,}2919})^{x} ⋅ 10^{3{,}96}\)

1p

○

\(y = 1{,}958...^{x} ⋅ 9120{,}108...\)
Dus \(y = 9\,120 ⋅ 1{,}96^{x} \text{.}\)

1p

3p

d

Schrijf de formule \(y = {}^{2}\!\log(2{,}2 x) - 1{,}9\) 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

d

\(y = {}^{2}\!\log(2{,}2 x) - 1{,}9\)
\(\text{ } = {}^{2}\!\log(2{,}2) + {}^{2}\!\log(x) - 1{,}9\)

1p

○

\(\text{ } = {}^{2}\!\log(2{,}2) - 1{,}9 + {{}^{5}\!\log(x) \over {}^{5}\!\log(2)}\)
\(\text{ } = {}^{2}\!\log(2{,}2) - 1{,}9 + {1 \over {}^{5}\!\log(2)} ⋅ {}^{5}\!\log(x)\)

1p

○

\(\text{ } = 1{,}137... - 1{,}9 + {1 \over 0{,}430...} ⋅ {}^{5}\!\log(x)\)
\(\text{ } = -0{,}762... + 2{,}321... ⋅ {}^{5}\!\log(x)\)
Dus \(y = -0{,}76 + 2{,}32 ⋅ {}^{5}\!\log(x) \text{.}\)

1p
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