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

\(y=24+3⋅{}^{9}\!\log(4x+6)\)

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

○

\(y=24+3⋅{}^{9}\!\log(4x+6)\)
\(3⋅{}^{9}\!\log(4x+6)=y-24\)
\({}^{9}\!\log(4x+6)=\frac{1}{3}y-8\)

1p

○

\(4x+6=9^{\frac{1}{3}y-8}\)

1p

○

\(4x=9^{\frac{1}{3}y-8}-6\)
\(x=\frac{1}{4}⋅9^{\frac{1}{3}y-8}-1\frac{1}{2}\)

1p

opgave 2

Herleid tot de gevraagde vorm.

3p

a

Schrijf de formule \(y=3\,400⋅0{,}8^x\) 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

\(y=3\,400⋅0{,}8^x\)
\(\log(y)=\log(3\,400⋅0{,}8^x)\)
\(\log(y)=\log(3\,400)+\log(0{,}8^x)\)

1p

○

\(\log(y)=\log(3\,400)+x⋅\log(0{,}8)\)

1p

○

\(\log(y)=3{,}531...+x⋅-0{,}09691...\)
Dus \(\log(y)=-0{,}0969x+3{,}53\)

1p

3p

b

Schrijf de formule \(y=2\,100⋅0{,}81^{4x+2}\) 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

\(y=2\,100⋅0{,}81^{4x+2}\)
\(\log(y)=\log(2\,100⋅0{,}81^{4x+2})\)
\(\log(y)=\log(2\,100)+\log(0{,}81^{4x+2})\)

1p

○

\(\log(y)=\log(2\,100)+(4x+2)⋅\log(0{,}81)\)
\(\log(y)=\log(2\,100)+4x⋅\log(0{,}81)+2⋅\log(0{,}81)\)

1p

○

\(\log(y)=3{,}322...+4x⋅-0{,}09151...+2⋅-0{,}09151...\)
\(\log(y)=3{,}322...-0{,}36605...⋅x-0{,}18302...\)
Dus \(\log(y)=-0{,}3661x+3{,}14\)

1p

3p

c

Schrijf de formule \(\log(y)=0{,}9594x+3{,}45\) 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

\(\log(y)=0{,}9594x+3{,}45\)
\(y=10^{0{,}9594x+3{,}45}\)

1p

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\(y=10^{0{,}9594x}⋅10^{3{,}45}\)
\(y=(10^{0{,}9594})^x⋅10^{3{,}45}\)

1p

○

\(y=9{,}107...^x⋅2818{,}382...\)
Dus \(y=2\,818⋅9{,}11^x\text{.}\)

1p

3p

d

Schrijf de formule \(y={}^{2}\!\log(2{,}7x)+0{,}4\) 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

\(y={}^{2}\!\log(2{,}7x)+0{,}4\)
\(\text{ }={}^{2}\!\log(2{,}7)+{}^{2}\!\log(x)+0{,}4\)

1p

○

\(\text{ }={}^{2}\!\log(2{,}7)+0{,}4+{{}^{3}\!\log(x) \over {}^{3}\!\log(2)}\)
\(\text{ }={}^{2}\!\log(2{,}7)+0{,}4+{1 \over {}^{3}\!\log(2)}⋅{}^{3}\!\log(x)\)

1p

○

\(\text{ }=1{,}432...+0{,}4+{1 \over 0{,}630...}⋅{}^{3}\!\log(x)\)
\(\text{ }=1{,}832...+1{,}584...⋅{}^{3}\!\log(x)\)
Dus \(y=1{,}83+1{,}58⋅{}^{3}\!\log(x)\text{.}\)

1p
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