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

\(y = 16 + 2 ⋅ {}^{9}\!\log(5 x + 7)\)

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

○

\(y = 16 + 2 ⋅ {}^{9}\!\log(5 x + 7)\)
\(2 ⋅ {}^{9}\!\log(5 x + 7) = y - 16\)
\({}^{9}\!\log(5 x + 7) = \frac{1}{2} y - 8\)

1p

○

\(5 x + 7 = 9^{\frac{1}{2} y - 8}\)

1p

○

\(5 x = 9^{\frac{1}{2} y - 8} - 7\)
\(x = \frac{1}{5} ⋅ 9^{\frac{1}{2} y - 8} - 1\frac{2}{5}\)

1p

opgave 2

Herleid tot de gevraagde vorm.

3p

a

Schrijf de formule \(y = 2\,700 ⋅ 0{,}71^{x}\) 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

\(y = 2\,700 ⋅ 0{,}71^{x}\)
\(\log(y) = \log(2\,700 ⋅ 0{,}71^{x})\)
\(\log(y) = \log(2\,700) + \log(0{,}71^{x})\)

1p

○

\(\log(y) = \log(2\,700) + x ⋅ \log(0{,}71)\)

1p

○

\(\log(y) = 3{,}431... + x ⋅ -0{,}14874...\)
Dus \(\log(y) = -0{,}1487 x + 3{,}43\)

1p

3p

b

Schrijf de formule \(y = 4\,500 ⋅ 1{,}1^{6 x + 3}\) 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

\(y = 4\,500 ⋅ 1{,}1^{6 x + 3}\)
\(\log(y) = \log(4\,500 ⋅ 1{,}1^{6 x + 3})\)
\(\log(y) = \log(4\,500) + \log(1{,}1^{6 x + 3})\)

1p

○

\(\log(y) = \log(4\,500) + (6 x + 3) ⋅ \log(1{,}1)\)
\(\log(y) = \log(4\,500) + 6 x ⋅ \log(1{,}1) + 3 ⋅ \log(1{,}1)\)

1p

○

\(\log(y) = 3{,}653... + 6 x ⋅ 0{,}04139... + 3 ⋅ 0{,}04139...\)
\(\log(y) = 3{,}653... + 0{,}24835... ⋅ x + 0{,}12417...\)
Dus \(\log(y) = 0{,}2484 x + 3{,}78\)

1p

3p

c

Schrijf de formule \(\log(y) = 0{,}4771 x + 2{,}78\) 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

\(\log(y) = 0{,}4771 x + 2{,}78\)
\(y = 10^{0{,}4771 x + 2{,}78}\)

1p

○

\(y = 10^{0{,}4771 x} ⋅ 10^{2{,}78}\)
\(y = (10^{0{,}4771})^{x} ⋅ 10^{2{,}78}\)

1p

○

\(y = 2{,}999...^{x} ⋅ 602{,}559...\)
Dus \(y = 603 ⋅ 3{,}00^{x} \text{.}\)

1p

3p

d

Schrijf de formule \(y = {}^{5}\!\log(1{,}6 x) + 1{,}3\) 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

\(y = {}^{5}\!\log(1{,}6 x) + 1{,}3\)
\(\text{ } = {}^{5}\!\log(1{,}6) + {}^{5}\!\log(x) + 1{,}3\)

1p

○

\(\text{ } = {}^{5}\!\log(1{,}6) + 1{,}3 + {{}^{2}\!\log(x) \over {}^{2}\!\log(5)}\)
\(\text{ } = {}^{5}\!\log(1{,}6) + 1{,}3 + {1 \over {}^{2}\!\log(5)} ⋅ {}^{2}\!\log(x)\)

1p

○

\(\text{ } = 0{,}292... + 1{,}3 + {1 \over 2{,}321...} ⋅ {}^{2}\!\log(x)\)
\(\text{ } = 1{,}592... + 0{,}430... ⋅ {}^{2}\!\log(x)\)
Dus \(y = 1{,}59 + 0{,}43 ⋅ {}^{2}\!\log(x) \text{.}\)

1p
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