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

\(y = 36 + 4 ⋅ {}^{2}\!\log(6 x + 8)\)

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

○

\(y = 36 + 4 ⋅ {}^{2}\!\log(6 x + 8)\)
\(4 ⋅ {}^{2}\!\log(6 x + 8) = y - 36\)
\({}^{2}\!\log(6 x + 8) = \frac{1}{4} y - 9\)

1p

○

\(6 x + 8 = 2^{\frac{1}{4} y - 9}\)

1p

○

\(6 x = 2^{\frac{1}{4} y - 9} - 8\)
\(x = \frac{1}{6} ⋅ 2^{\frac{1}{4} y - 9} - 1\frac{1}{3}\)

1p

opgave 2

Herleid tot de gevraagde vorm.

3p

a

Schrijf de formule \(y = 4\,800 ⋅ 1{,}25^{x}\) 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

\(y = 4\,800 ⋅ 1{,}25^{x}\)
\(\log(y) = \log(4\,800 ⋅ 1{,}25^{x})\)
\(\log(y) = \log(4\,800) + \log(1{,}25^{x})\)

1p

○

\(\log(y) = \log(4\,800) + x ⋅ \log(1{,}25)\)

1p

○

\(\log(y) = 3{,}681... + x ⋅ 0{,}09691...\)
Dus \(\log(y) = 0{,}0969 x + 3{,}68\)

1p

3p

b

Schrijf de formule \(y = 6\,900 ⋅ 0{,}89^{4 x + 6}\) 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

\(y = 6\,900 ⋅ 0{,}89^{4 x + 6}\)
\(\log(y) = \log(6\,900 ⋅ 0{,}89^{4 x + 6})\)
\(\log(y) = \log(6\,900) + \log(0{,}89^{4 x + 6})\)

1p

○

\(\log(y) = \log(6\,900) + (4 x + 6) ⋅ \log(0{,}89)\)
\(\log(y) = \log(6\,900) + 4 x ⋅ \log(0{,}89) + 6 ⋅ \log(0{,}89)\)

1p

○

\(\log(y) = 3{,}838... + 4 x ⋅ -0{,}05060... + 6 ⋅ -0{,}05060...\)
\(\log(y) = 3{,}838... - 0{,}20243... ⋅ x - 0{,}30365...\)
Dus \(\log(y) = -0{,}2024 x + 3{,}54\)

1p

3p

c

Schrijf de formule \(\log(y) = -0{,}9171 x + 2{,}49\) 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

\(\log(y) = -0{,}9171 x + 2{,}49\)
\(y = 10^{-0{,}9171 x + 2{,}49}\)

1p

○

\(y = 10^{-0{,}9171 x} ⋅ 10^{2{,}49}\)
\(y = (10^{-0{,}9171})^{x} ⋅ 10^{2{,}49}\)

1p

○

\(y = 0{,}121...^{x} ⋅ 309{,}029...\)
Dus \(y = 309 ⋅ 0{,}12^{x} \text{.}\)

1p

3p

d

Schrijf de formule \(y = {}^{2}\!\log(2{,}4 x) - 2{,}1\) in de vorm \(y = a + b ⋅ {}^{4}\!\log(x) \text{.}\)
Geef \(a\) en \(b\) in twee decimalen.

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

d

\(y = {}^{2}\!\log(2{,}4 x) - 2{,}1\)
\(\text{ } = {}^{2}\!\log(2{,}4) + {}^{2}\!\log(x) - 2{,}1\)

1p

○

\(\text{ } = {}^{2}\!\log(2{,}4) - 2{,}1 + {{}^{4}\!\log(x) \over {}^{4}\!\log(2)}\)
\(\text{ } = {}^{2}\!\log(2{,}4) - 2{,}1 + {1 \over {}^{4}\!\log(2)} ⋅ {}^{4}\!\log(x)\)

1p

○

\(\text{ } = 1{,}263... - 2{,}1 + {1 \over 0{,}5} ⋅ {}^{4}\!\log(x)\)
\(\text{ } = -0{,}836... + 2 ⋅ {}^{4}\!\log(x)\)
Dus \(y = -0{,}84 + 2{,}00 ⋅ {}^{4}\!\log(x) \text{.}\)

1p
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