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

\(y = 14 + 2 ⋅ {}^{9}\!\log(6 x - 4)\)

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

○

\(y = 14 + 2 ⋅ {}^{9}\!\log(6 x - 4)\)
\(2 ⋅ {}^{9}\!\log(6 x - 4) = y - 14\)
\({}^{9}\!\log(6 x - 4) = \frac{1}{2} y - 7\)

1p

○

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

1p

○

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

1p

opgave 2

Herleid tot de gevraagde vorm.

3p

a

Schrijf de formule \(y = 4\,300 ⋅ 1{,}15^{x}\) 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

\(y = 4\,300 ⋅ 1{,}15^{x}\)
\(\log(y) = \log(4\,300 ⋅ 1{,}15^{x})\)
\(\log(y) = \log(4\,300) + \log(1{,}15^{x})\)

1p

○

\(\log(y) = \log(4\,300) + x ⋅ \log(1{,}15)\)

1p

○

\(\log(y) = 3{,}633... + x ⋅ 0{,}06069...\)
Dus \(\log(y) = 0{,}0607 x + 3{,}63\)

1p

3p

b

Schrijf de formule \(y = 8\,900 ⋅ 0{,}78^{6 x + 5}\) 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

\(y = 8\,900 ⋅ 0{,}78^{6 x + 5}\)
\(\log(y) = \log(8\,900 ⋅ 0{,}78^{6 x + 5})\)
\(\log(y) = \log(8\,900) + \log(0{,}78^{6 x + 5})\)

1p

○

\(\log(y) = \log(8\,900) + (6 x + 5) ⋅ \log(0{,}78)\)
\(\log(y) = \log(8\,900) + 6 x ⋅ \log(0{,}78) + 5 ⋅ \log(0{,}78)\)

1p

○

\(\log(y) = 3{,}949... + 6 x ⋅ -0{,}10790... + 5 ⋅ -0{,}10790...\)
\(\log(y) = 3{,}949... - 0{,}64743... ⋅ x - 0{,}53952...\)
Dus \(\log(y) = -0{,}6474 x + 3{,}41\)

1p

3p

c

Schrijf de formule \(\log(y) = 0{,}1027 x + 3{,}53\) 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

\(\log(y) = 0{,}1027 x + 3{,}53\)
\(y = 10^{0{,}1027 x + 3{,}53}\)

1p

○

\(y = 10^{0{,}1027 x} ⋅ 10^{3{,}53}\)
\(y = (10^{0{,}1027})^{x} ⋅ 10^{3{,}53}\)

1p

○

\(y = 1{,}266...^{x} ⋅ 3388{,}441...\)
Dus \(y = 3\,388 ⋅ 1{,}27^{x} \text{.}\)

1p

3p

d

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

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

1p

○

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

1p

○

\(\text{ } = 0{,}086... + 2{,}3 + {1 \over 0{,}792...} ⋅ {}^{4}\!\log(x)\)
\(\text{ } = 2{,}386... + 1{,}261... ⋅ {}^{4}\!\log(x)\)
Dus \(y = 2{,}39 + 1{,}26 ⋅ {}^{4}\!\log(x) \text{.}\)

1p
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