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

\(y = 36 + 4 ⋅ {}^{7}\!\log(6 x - 3)\)

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

○

\(y = 36 + 4 ⋅ {}^{7}\!\log(6 x - 3)\)
\(4 ⋅ {}^{7}\!\log(6 x - 3) = y - 36\)
\({}^{7}\!\log(6 x - 3) = \frac{1}{4} y - 9\)

1p

○

\(6 x - 3 = 7^{\frac{1}{4} y - 9}\)

1p

○

\(6 x = 7^{\frac{1}{4} y - 9} + 3\)
\(x = \frac{1}{6} ⋅ 7^{\frac{1}{4} y - 9} + \frac{1}{2}\)

1p

opgave 2

Herleid tot de gevraagde vorm.

3p

a

Schrijf de formule \(y = 9\,500 ⋅ 1{,}15^{x}\) 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

\(y = 9\,500 ⋅ 1{,}15^{x}\)
\(\log(y) = \log(9\,500 ⋅ 1{,}15^{x})\)
\(\log(y) = \log(9\,500) + \log(1{,}15^{x})\)

1p

○

\(\log(y) = \log(9\,500) + x ⋅ \log(1{,}15)\)

1p

○

\(\log(y) = 3{,}977... + x ⋅ 0{,}06069...\)
Dus \(\log(y) = 0{,}0607 x + 3{,}98\)

1p

3p

b

Schrijf de formule \(y = 5\,700 ⋅ 0{,}77^{6 x + 5}\) 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

\(y = 5\,700 ⋅ 0{,}77^{6 x + 5}\)
\(\log(y) = \log(5\,700 ⋅ 0{,}77^{6 x + 5})\)
\(\log(y) = \log(5\,700) + \log(0{,}77^{6 x + 5})\)

1p

○

\(\log(y) = \log(5\,700) + (6 x + 5) ⋅ \log(0{,}77)\)
\(\log(y) = \log(5\,700) + 6 x ⋅ \log(0{,}77) + 5 ⋅ \log(0{,}77)\)

1p

○

\(\log(y) = 3{,}755... + 6 x ⋅ -0{,}11350... + 5 ⋅ -0{,}11350...\)
\(\log(y) = 3{,}755... - 0{,}68105... ⋅ x - 0{,}56754...\)
Dus \(\log(y) = -0{,}6811 x + 3{,}19\)

1p

3p

c

Schrijf de formule \(\log(y) = -0{,}0508 x + 3{,}68\) 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

\(\log(y) = -0{,}0508 x + 3{,}68\)
\(y = 10^{-0{,}0508 x + 3{,}68}\)

1p

○

\(y = 10^{-0{,}0508 x} ⋅ 10^{3{,}68}\)
\(y = (10^{-0{,}0508})^{x} ⋅ 10^{3{,}68}\)

1p

○

\(y = 0{,}889...^{x} ⋅ 4786{,}300...\)
Dus \(y = 4\,786 ⋅ 0{,}89^{x} \text{.}\)

1p

3p

d

Schrijf de formule \(y = {}^{3}\!\log(1{,}1 x) + 2{,}5\) 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

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

1p

○

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

1p

○

\(\text{ } = 0{,}086... + 2{,}5 + {1 \over 0{,}682...} ⋅ {}^{5}\!\log(x)\)
\(\text{ } = 2{,}586... + 1{,}464... ⋅ {}^{5}\!\log(x)\)
Dus \(y = 2{,}59 + 1{,}46 ⋅ {}^{5}\!\log(x) \text{.}\)

1p
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