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

\(y = 6 + 2 ⋅ {}^{5}\!\log(9 x + 6)\)

Vrijmaken
00kn - Logaritmische formules herleiden - basis - 0ms - dynamic variables

○

\(y = 6 + 2 ⋅ {}^{5}\!\log(9 x + 6)\)
\(2 ⋅ {}^{5}\!\log(9 x + 6) = y - 6\)
\({}^{5}\!\log(9 x + 6) = \frac{1}{2} y - 3\)

1p

○

\(9 x + 6 = 5^{\frac{1}{2} y - 3}\)

1p

○

\(9 x = 5^{\frac{1}{2} y - 3} - 6\)
\(x = \frac{1}{9} ⋅ 5^{\frac{1}{2} y - 3} - \frac{2}{3}\)

1p

opgave 2

Herleid tot de gevraagde vorm.

3p

a

Schrijf de formule \(y = 1\,700 ⋅ 0{,}78^{x}\) 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

\(y = 1\,700 ⋅ 0{,}78^{x}\)
\(\log(y) = \log(1\,700 ⋅ 0{,}78^{x})\)
\(\log(y) = \log(1\,700) + \log(0{,}78^{x})\)

1p

○

\(\log(y) = \log(1\,700) + x ⋅ \log(0{,}78)\)

1p

○

\(\log(y) = 3{,}230... + x ⋅ -0{,}10790...\)
Dus \(\log(y) = -0{,}1079 x + 3{,}23\)

1p

3p

b

Schrijf de formule \(y = 7\,900 ⋅ 1{,}23^{2 x + 5}\) 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 - 0ms - dynamic variables

b

\(y = 7\,900 ⋅ 1{,}23^{2 x + 5}\)
\(\log(y) = \log(7\,900 ⋅ 1{,}23^{2 x + 5})\)
\(\log(y) = \log(7\,900) + \log(1{,}23^{2 x + 5})\)

1p

○

\(\log(y) = \log(7\,900) + (2 x + 5) ⋅ \log(1{,}23)\)
\(\log(y) = \log(7\,900) + 2 x ⋅ \log(1{,}23) + 5 ⋅ \log(1{,}23)\)

1p

○

\(\log(y) = 3{,}897... + 2 x ⋅ 0{,}08990... + 5 ⋅ 0{,}08990...\)
\(\log(y) = 3{,}897... + 0{,}17981... ⋅ x + 0{,}44952...\)
Dus \(\log(y) = 0{,}1798 x + 4{,}35\)

1p

3p

c

Schrijf de formule \(\log(y) = 0{,}9286 x + 3{,}51\) 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

\(\log(y) = 0{,}9286 x + 3{,}51\)
\(y = 10^{0{,}9286 x + 3{,}51}\)

1p

○

\(y = 10^{0{,}9286 x} ⋅ 10^{3{,}51}\)
\(y = (10^{0{,}9286})^{x} ⋅ 10^{3{,}51}\)

1p

○

\(y = 8{,}483...^{x} ⋅ 3235{,}936...\)
Dus \(y = 3\,236 ⋅ 8{,}48^{x} \text{.}\)

1p

3p

d

Schrijf de formule \(y = {}^{3}\!\log(1{,}6 x) - 1{,}1\) in de vorm \(y = a + b ⋅ {}^{2}\!\log(x) \text{.}\)
Geef \(a\) en \(b\) in twee decimalen.

Herleiden (6)
00l2 - Logaritmische formules herleiden - basis - 0ms - dynamic variables

d

\(y = {}^{3}\!\log(1{,}6 x) - 1{,}1\)
\(\text{ } = {}^{3}\!\log(1{,}6) + {}^{3}\!\log(x) - 1{,}1\)

1p

○

\(\text{ } = {}^{3}\!\log(1{,}6) - 1{,}1 + {{}^{2}\!\log(x) \over {}^{2}\!\log(3)}\)
\(\text{ } = {}^{3}\!\log(1{,}6) - 1{,}1 + {1 \over {}^{2}\!\log(3)} ⋅ {}^{2}\!\log(x)\)

1p

○

\(\text{ } = 0{,}427... - 1{,}1 + {1 \over 1{,}584...} ⋅ {}^{2}\!\log(x)\)
\(\text{ } = -0{,}672... + 0{,}630... ⋅ {}^{2}\!\log(x)\)
Dus \(y = -0{,}67 + 0{,}63 ⋅ {}^{2}\!\log(x) \text{.}\)

1p
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