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

\(y = 2 + 2 ⋅ {}^{9}\!\log(4 x + 5)\)

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

○

\(y = 2 + 2 ⋅ {}^{9}\!\log(4 x + 5)\)
\(2 ⋅ {}^{9}\!\log(4 x + 5) = y - 2\)
\({}^{9}\!\log(4 x + 5) = \frac{1}{2} y - 1\)

1p

○

\(4 x + 5 = 9^{\frac{1}{2} y - 1}\)

1p

○

\(4 x = 9^{\frac{1}{2} y - 1} - 5\)
\(x = \frac{1}{4} ⋅ 9^{\frac{1}{2} y - 1} - 1\frac{1}{4}\)

1p

opgave 2

Herleid tot de gevraagde vorm.

3p

a

Schrijf de formule \(y = 5\,000 ⋅ 0{,}76^{x}\) 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

\(y = 5\,000 ⋅ 0{,}76^{x}\)
\(\log(y) = \log(5\,000 ⋅ 0{,}76^{x})\)
\(\log(y) = \log(5\,000) + \log(0{,}76^{x})\)

1p

○

\(\log(y) = \log(5\,000) + x ⋅ \log(0{,}76)\)

1p

○

\(\log(y) = 3{,}698... + x ⋅ -0{,}11918...\)
Dus \(\log(y) = -0{,}1192 x + 3{,}70\)

1p

3p

b

Schrijf de formule \(y = 1\,000 ⋅ 0{,}7^{3 x + 2}\) 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

\(y = 1\,000 ⋅ 0{,}7^{3 x + 2}\)
\(\log(y) = \log(1\,000 ⋅ 0{,}7^{3 x + 2})\)
\(\log(y) = \log(1\,000) + \log(0{,}7^{3 x + 2})\)

1p

○

\(\log(y) = \log(1\,000) + (3 x + 2) ⋅ \log(0{,}7)\)
\(\log(y) = \log(1\,000) + 3 x ⋅ \log(0{,}7) + 2 ⋅ \log(0{,}7)\)

1p

○

\(\log(y) = 3 + 3 x ⋅ -0{,}15490... + 2 ⋅ -0{,}15490...\)
\(\log(y) = 3 - 0{,}46470... ⋅ x - 0{,}30980...\)
Dus \(\log(y) = -0{,}4647 x + 2{,}69\)

1p

3p

c

Schrijf de formule \(\log(y) = 0{,}9931 x + 2{,}96\) 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

\(\log(y) = 0{,}9931 x + 2{,}96\)
\(y = 10^{0{,}9931 x + 2{,}96}\)

1p

○

\(y = 10^{0{,}9931 x} ⋅ 10^{2{,}96}\)
\(y = (10^{0{,}9931})^{x} ⋅ 10^{2{,}96}\)

1p

○

\(y = 9{,}842...^{x} ⋅ 912{,}010...\)
Dus \(y = 912 ⋅ 9{,}84^{x} \text{.}\)

1p

3p

d

Schrijf de formule \(y = {}^{4}\!\log(1{,}6 x) - 2{,}1\) in de vorm \(y = a + b ⋅ {}^{3}\!\log(x) \text{.}\)
Geef \(a\) en \(b\) in twee decimalen.

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

d

\(y = {}^{4}\!\log(1{,}6 x) - 2{,}1\)
\(\text{ } = {}^{4}\!\log(1{,}6) + {}^{4}\!\log(x) - 2{,}1\)

1p

○

\(\text{ } = {}^{4}\!\log(1{,}6) - 2{,}1 + {{}^{3}\!\log(x) \over {}^{3}\!\log(4)}\)
\(\text{ } = {}^{4}\!\log(1{,}6) - 2{,}1 + {1 \over {}^{3}\!\log(4)} ⋅ {}^{3}\!\log(x)\)

1p

○

\(\text{ } = 0{,}339... - 2{,}1 + {1 \over 1{,}261...} ⋅ {}^{3}\!\log(x)\)
\(\text{ } = -1{,}760... + 0{,}792... ⋅ {}^{3}\!\log(x)\)
Dus \(y = -1{,}76 + 0{,}79 ⋅ {}^{3}\!\log(x) \text{.}\)

1p
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