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

\(y=20+4⋅{}^{2}\!\log(6x+1)\)

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

○

\(y=20+4⋅{}^{2}\!\log(6x+1)\)
\(4⋅{}^{2}\!\log(6x+1)=y-20\)
\({}^{2}\!\log(6x+1)=\frac{1}{4}y-5\)

1p

○

\(6x+1=2^{\frac{1}{4}y-5}\)

1p

○

\(6x=2^{\frac{1}{4}y-5}-1\)
\(x=\frac{1}{6}⋅2^{\frac{1}{4}y-5}-\frac{1}{6}\)

1p

opgave 2

Herleid tot de gevraagde vorm.

3p

a

Schrijf de formule \(y=8\,200⋅0{,}88^x\) 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 - 0ms - dynamic variables

a

\(y=8\,200⋅0{,}88^x\)
\(\log(y)=\log(8\,200⋅0{,}88^x)\)
\(\log(y)=\log(8\,200)+\log(0{,}88^x)\)

1p

○

\(\log(y)=\log(8\,200)+x⋅\log(0{,}88)\)

1p

○

\(\log(y)=3{,}913...+x⋅-0{,}05551...\)
Dus \(\log(y)=-0{,}0555x+3{,}91\)

1p

3p

b

Schrijf de formule \(y=2\,800⋅1{,}1^{4x+5}\) 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

\(y=2\,800⋅1{,}1^{4x+5}\)
\(\log(y)=\log(2\,800⋅1{,}1^{4x+5})\)
\(\log(y)=\log(2\,800)+\log(1{,}1^{4x+5})\)

1p

○

\(\log(y)=\log(2\,800)+(4x+5)⋅\log(1{,}1)\)
\(\log(y)=\log(2\,800)+4x⋅\log(1{,}1)+5⋅\log(1{,}1)\)

1p

○

\(\log(y)=3{,}447...+4x⋅0{,}04139...+5⋅0{,}04139...\)
\(\log(y)=3{,}447...+0{,}16557...⋅x+0{,}20696...\)
Dus \(\log(y)=0{,}1656x+3{,}65\)

1p

3p

c

Schrijf de formule \(\log(y)=0{,}1236x+1{,}48\) 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

\(\log(y)=0{,}1236x+1{,}48\)
\(y=10^{0{,}1236x+1{,}48}\)

1p

○

\(y=10^{0{,}1236x}⋅10^{1{,}48}\)
\(y=(10^{0{,}1236})^x⋅10^{1{,}48}\)

1p

○

\(y=1{,}329...^x⋅30{,}199...\)
Dus \(y=30⋅1{,}33^x\text{.}\)

1p

3p

d

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

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

1p

○

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

1p

○

\(\text{ }=0{,}460...-2{,}3+{1 \over 1{,}160...}⋅{}^{4}\!\log(x)\)
\(\text{ }=-1{,}839...+0{,}861...⋅{}^{4}\!\log(x)\)
Dus \(y=-1{,}84+0{,}86⋅{}^{4}\!\log(x)\text{.}\)

1p
"