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

\(y = 6 + 2 ⋅ {}^{6}\!\log(7 x - 9)\)

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

○

\(y = 6 + 2 ⋅ {}^{6}\!\log(7 x - 9)\)
\(2 ⋅ {}^{6}\!\log(7 x - 9) = y - 6\)
\({}^{6}\!\log(7 x - 9) = \frac{1}{2} y - 3\)

1p

○

\(7 x - 9 = 6^{\frac{1}{2} y - 3}\)

1p

○

\(7 x = 6^{\frac{1}{2} y - 3} + 9\)
\(x = \frac{1}{7} ⋅ 6^{\frac{1}{2} y - 3} + 1\frac{2}{7}\)

1p

opgave 2

Herleid tot de gevraagde vorm.

3p

a

Schrijf de formule \(y = 1\,100 ⋅ 0{,}93^{x}\) 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

\(y = 1\,100 ⋅ 0{,}93^{x}\)
\(\log(y) = \log(1\,100 ⋅ 0{,}93^{x})\)
\(\log(y) = \log(1\,100) + \log(0{,}93^{x})\)

1p

○

\(\log(y) = \log(1\,100) + x ⋅ \log(0{,}93)\)

1p

○

\(\log(y) = 3{,}041... + x ⋅ -0{,}03151...\)
Dus \(\log(y) = -0{,}0315 x + 3{,}04\)

1p

3p

b

Schrijf de formule \(y = 3\,600 ⋅ 0{,}72^{2 x + 5}\) 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

\(y = 3\,600 ⋅ 0{,}72^{2 x + 5}\)
\(\log(y) = \log(3\,600 ⋅ 0{,}72^{2 x + 5})\)
\(\log(y) = \log(3\,600) + \log(0{,}72^{2 x + 5})\)

1p

○

\(\log(y) = \log(3\,600) + (2 x + 5) ⋅ \log(0{,}72)\)
\(\log(y) = \log(3\,600) + 2 x ⋅ \log(0{,}72) + 5 ⋅ \log(0{,}72)\)

1p

○

\(\log(y) = 3{,}556... + 2 x ⋅ -0{,}14266... + 5 ⋅ -0{,}14266...\)
\(\log(y) = 3{,}556... - 0{,}28533... ⋅ x - 0{,}71333...\)
Dus \(\log(y) = -0{,}2853 x + 2{,}84\)

1p

3p

c

Schrijf de formule \(\log(y) = 0{,}7903 x + 3{,}53\) 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

\(\log(y) = 0{,}7903 x + 3{,}53\)
\(y = 10^{0{,}7903 x + 3{,}53}\)

1p

○

\(y = 10^{0{,}7903 x} ⋅ 10^{3{,}53}\)
\(y = (10^{0{,}7903})^{x} ⋅ 10^{3{,}53}\)

1p

○

\(y = 6{,}170...^{x} ⋅ 3388{,}441...\)
Dus \(y = 3\,388 ⋅ 6{,}17^{x} \text{.}\)

1p

3p

d

Schrijf de formule \(y = {}^{4}\!\log(2{,}9 x) - 2{,}9\) 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

\(y = {}^{4}\!\log(2{,}9 x) - 2{,}9\)
\(\text{ } = {}^{4}\!\log(2{,}9) + {}^{4}\!\log(x) - 2{,}9\)

1p

○

\(\text{ } = {}^{4}\!\log(2{,}9) - 2{,}9 + {{}^{5}\!\log(x) \over {}^{5}\!\log(4)}\)
\(\text{ } = {}^{4}\!\log(2{,}9) - 2{,}9 + {1 \over {}^{5}\!\log(4)} ⋅ {}^{5}\!\log(x)\)

1p

○

\(\text{ } = 0{,}768... - 2{,}9 + {1 \over 0{,}861...} ⋅ {}^{5}\!\log(x)\)
\(\text{ } = -2{,}131... + 1{,}160... ⋅ {}^{5}\!\log(x)\)
Dus \(y = -2{,}13 + 1{,}16 ⋅ {}^{5}\!\log(x) \text{.}\)

1p
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