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

\(y=24+3⋅{}^{4}\!\log(2x-5)\)

Vrijmaken
00kn - Logaritmische formules herleiden - basis - 2ms - dynamic variables

○

\(y=24+3⋅{}^{4}\!\log(2x-5)\)
\(3⋅{}^{4}\!\log(2x-5)=y-24\)
\({}^{4}\!\log(2x-5)=\frac{1}{3}y-8\)

1p

○

\(2x-5=4^{\frac{1}{3}y-8}\)

1p

○

\(2x=4^{\frac{1}{3}y-8}+5\)
\(x=\frac{1}{2}⋅4^{\frac{1}{3}y-8}+2\frac{1}{2}\)

1p

opgave 2

Herleid tot de gevraagde vorm.

3p

a

Schrijf de formule \(y=9\,200⋅0{,}92^x\) in de vorm \(\log(y)=ax+b\text{.}\)
Geef \(a\) in vier decimalen en \(b\) in twee decimalen.

Herleiden (1)
00ko - Logaritmische formules herleiden - basis - 1ms - dynamic variables

a

\(y=9\,200⋅0{,}92^x\)
\(\log(y)=\log(9\,200⋅0{,}92^x)\)
\(\log(y)=\log(9\,200)+\log(0{,}92^x)\)

1p

○

\(\log(y)=\log(9\,200)+x⋅\log(0{,}92)\)

1p

○

\(\log(y)=3{,}963...+x⋅-0{,}03621...\)
Dus \(\log(y)=-0{,}0362x+3{,}96\)

1p

3p

b

Schrijf de formule \(y=7\,700⋅1{,}17^{4x+2}\) in de vorm \(\log(y)=ax+b\text{.}\)
Geef \(a\) in vier decimalen en \(b\) in twee decimalen.

Herleiden (2)
00kp - Logaritmische formules herleiden - basis - 1ms - dynamic variables

b

\(y=7\,700⋅1{,}17^{4x+2}\)
\(\log(y)=\log(7\,700⋅1{,}17^{4x+2})\)
\(\log(y)=\log(7\,700)+\log(1{,}17^{4x+2})\)

1p

○

\(\log(y)=\log(7\,700)+(4x+2)⋅\log(1{,}17)\)
\(\log(y)=\log(7\,700)+4x⋅\log(1{,}17)+2⋅\log(1{,}17)\)

1p

○

\(\log(y)=3{,}886...+4x⋅0{,}06818...+2⋅0{,}06818...\)
\(\log(y)=3{,}886...+0{,}27274...⋅x+0{,}13637...\)
Dus \(\log(y)=0{,}2727x+4{,}02\)

1p

3p

c

Schrijf de formule \(\log(y)=-0{,}6928x+1{,}19\) 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

\(\log(y)=-0{,}6928x+1{,}19\)
\(y=10^{-0{,}6928x+1{,}19}\)

1p

○

\(y=10^{-0{,}6928x}⋅10^{1{,}19}\)
\(y=(10^{-0{,}6928})^x⋅10^{1{,}19}\)

1p

○

\(y=0{,}202...^x⋅15{,}488...\)
Dus \(y=15⋅0{,}20^x\text{.}\)

1p

3p

d

Schrijf de formule \(y={}^{2}\!\log(2{,}9x)-2{,}1\) in de vorm \(y=a+b⋅{}^{3}\!\log(x)\text{.}\)
Geef \(a\) en \(b\) in twee decimalen.

Herleiden (6)
00l2 - Logaritmische formules herleiden - basis - 1ms - dynamic variables

d

\(y={}^{2}\!\log(2{,}9x)-2{,}1\)
\(\text{ }={}^{2}\!\log(2{,}9)+{}^{2}\!\log(x)-2{,}1\)

1p

○

\(\text{ }={}^{2}\!\log(2{,}9)-2{,}1+{{}^{3}\!\log(x) \over {}^{3}\!\log(2)}\)
\(\text{ }={}^{2}\!\log(2{,}9)-2{,}1+{1 \over {}^{3}\!\log(2)}⋅{}^{3}\!\log(x)\)

1p

○

\(\text{ }=1{,}536...-2{,}1+{1 \over 0{,}630...}⋅{}^{3}\!\log(x)\)
\(\text{ }=-0{,}563...+1{,}584...⋅{}^{3}\!\log(x)\)
Dus \(y=-0{,}56+1{,}58⋅{}^{3}\!\log(x)\text{.}\)

1p
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