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

\(y = 4 + 4 ⋅ {}^{8}\!\log(9 x + 3)\)

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

○

\(y = 4 + 4 ⋅ {}^{8}\!\log(9 x + 3)\)
\(4 ⋅ {}^{8}\!\log(9 x + 3) = y - 4\)
\({}^{8}\!\log(9 x + 3) = \frac{1}{4} y - 1\)

1p

○

\(9 x + 3 = 8^{\frac{1}{4} y - 1}\)

1p

○

\(9 x = 8^{\frac{1}{4} y - 1} - 3\)
\(x = \frac{1}{9} ⋅ 8^{\frac{1}{4} y - 1} - \frac{1}{3}\)

1p

opgave 2

Herleid tot de gevraagde vorm.

3p

a

Schrijf de formule \(y = 7\,900 ⋅ 0{,}89^{x}\) 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

\(y = 7\,900 ⋅ 0{,}89^{x}\)
\(\log(y) = \log(7\,900 ⋅ 0{,}89^{x})\)
\(\log(y) = \log(7\,900) + \log(0{,}89^{x})\)

1p

○

\(\log(y) = \log(7\,900) + x ⋅ \log(0{,}89)\)

1p

○

\(\log(y) = 3{,}897... + x ⋅ -0{,}05060...\)
Dus \(\log(y) = -0{,}0506 x + 3{,}90\)

1p

3p

b

Schrijf de formule \(y = 1\,100 ⋅ 0{,}74^{4 x + 2}\) 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 - 1ms - dynamic variables

b

\(y = 1\,100 ⋅ 0{,}74^{4 x + 2}\)
\(\log(y) = \log(1\,100 ⋅ 0{,}74^{4 x + 2})\)
\(\log(y) = \log(1\,100) + \log(0{,}74^{4 x + 2})\)

1p

○

\(\log(y) = \log(1\,100) + (4 x + 2) ⋅ \log(0{,}74)\)
\(\log(y) = \log(1\,100) + 4 x ⋅ \log(0{,}74) + 2 ⋅ \log(0{,}74)\)

1p

○

\(\log(y) = 3{,}041... + 4 x ⋅ -0{,}13076... + 2 ⋅ -0{,}13076...\)
\(\log(y) = 3{,}041... - 0{,}52307... ⋅ x - 0{,}26153...\)
Dus \(\log(y) = -0{,}5231 x + 2{,}78\)

1p

3p

c

Schrijf de formule \(\log(y) = -0{,}3051 x + 1{,}25\) 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

\(\log(y) = -0{,}3051 x + 1{,}25\)
\(y = 10^{-0{,}3051 x + 1{,}25}\)

1p

○

\(y = 10^{-0{,}3051 x} ⋅ 10^{1{,}25}\)
\(y = (10^{-0{,}3051})^{x} ⋅ 10^{1{,}25}\)

1p

○

\(y = 0{,}495...^{x} ⋅ 17{,}782...\)
Dus \(y = 18 ⋅ 0{,}50^{x} \text{.}\)

1p

3p

d

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

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

1p

○

\(\text{ } = {}^{3}\!\log(2{,}3) + 1{,}5 + {{}^{4}\!\log(x) \over {}^{4}\!\log(3)}\)
\(\text{ } = {}^{3}\!\log(2{,}3) + 1{,}5 + {1 \over {}^{4}\!\log(3)} ⋅ {}^{4}\!\log(x)\)

1p

○

\(\text{ } = 0{,}758... + 1{,}5 + {1 \over 0{,}792...} ⋅ {}^{4}\!\log(x)\)
\(\text{ } = 2{,}258... + 1{,}261... ⋅ {}^{4}\!\log(x)\)
Dus \(y = 2{,}26 + 1{,}26 ⋅ {}^{4}\!\log(x) \text{.}\)

1p
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