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

\(y=8+4⋅{}^{8}\!\log(7x+9)\)

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

○

\(y=8+4⋅{}^{8}\!\log(7x+9)\)
\(4⋅{}^{8}\!\log(7x+9)=y-8\)
\({}^{8}\!\log(7x+9)=\frac{1}{4}y-2\)

1p

○

\(7x+9=8^{\frac{1}{4}y-2}\)

1p

○

\(7x=8^{\frac{1}{4}y-2}-9\)
\(x=\frac{1}{7}⋅8^{\frac{1}{4}y-2}-1\frac{2}{7}\)

1p

opgave 2

Herleid tot de gevraagde vorm.

3p

a

Schrijf de formule \(y=3\,100⋅1{,}2^x\) 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

\(y=3\,100⋅1{,}2^x\)
\(\log(y)=\log(3\,100⋅1{,}2^x)\)
\(\log(y)=\log(3\,100)+\log(1{,}2^x)\)

1p

○

\(\log(y)=\log(3\,100)+x⋅\log(1{,}2)\)

1p

○

\(\log(y)=3{,}491...+x⋅0{,}07918...\)
Dus \(\log(y)=0{,}0792x+3{,}49\)

1p

3p

b

Schrijf de formule \(y=8\,400⋅1{,}2^{4x+6}\) 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

\(y=8\,400⋅1{,}2^{4x+6}\)
\(\log(y)=\log(8\,400⋅1{,}2^{4x+6})\)
\(\log(y)=\log(8\,400)+\log(1{,}2^{4x+6})\)

1p

○

\(\log(y)=\log(8\,400)+(4x+6)⋅\log(1{,}2)\)
\(\log(y)=\log(8\,400)+4x⋅\log(1{,}2)+6⋅\log(1{,}2)\)

1p

○

\(\log(y)=3{,}924...+4x⋅0{,}07918...+6⋅0{,}07918...\)
\(\log(y)=3{,}924...+0{,}31672...⋅x+0{,}47508...\)
Dus \(\log(y)=0{,}3167x+4{,}40\)

1p

3p

c

Schrijf de formule \(\log(y)=-0{,}3409x+2{,}43\) 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

\(\log(y)=-0{,}3409x+2{,}43\)
\(y=10^{-0{,}3409x+2{,}43}\)

1p

○

\(y=10^{-0{,}3409x}⋅10^{2{,}43}\)
\(y=(10^{-0{,}3409})^x⋅10^{2{,}43}\)

1p

○

\(y=0{,}456...^x⋅269{,}153...\)
Dus \(y=269⋅0{,}46^x\text{.}\)

1p

3p

d

Schrijf de formule \(y={}^{5}\!\log(1{,}7x)-1{,}2\) in de vorm \(y=a+b⋅{}^{2}\!\log(x)\text{.}\)
Geef \(a\) en \(b\) in twee decimalen.

Herleiden (6)
00l2 - Logaritmische formules herleiden - basis - 0ms - dynamic variables

d

\(y={}^{5}\!\log(1{,}7x)-1{,}2\)
\(\text{ }={}^{5}\!\log(1{,}7)+{}^{5}\!\log(x)-1{,}2\)

1p

○

\(\text{ }={}^{5}\!\log(1{,}7)-1{,}2+{{}^{2}\!\log(x) \over {}^{2}\!\log(5)}\)
\(\text{ }={}^{5}\!\log(1{,}7)-1{,}2+{1 \over {}^{2}\!\log(5)}⋅{}^{2}\!\log(x)\)

1p

○

\(\text{ }=0{,}329...-1{,}2+{1 \over 2{,}321...}⋅{}^{2}\!\log(x)\)
\(\text{ }=-0{,}870...+0{,}430...⋅{}^{2}\!\log(x)\)
Dus \(y=-0{,}87+0{,}43⋅{}^{2}\!\log(x)\text{.}\)

1p
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