Getal & Ruimte (12e editie) - havo wiskunde B

'Logaritmische formules herleiden'.

havo wiskunde B	9.2 Werken met logaritmen
Logaritmische formules herleiden (1)	opgave 1 Druk \(x\) uit in \(y\text{.}\) 3p \(y=32+4⋅{}^{5}\!\log(6x+1)\) Vrijmaken 00kn - Logaritmische formules herleiden - basis - 1ms - dynamic variables ○ \(y=32+4⋅{}^{5}\!\log(6x+1)\) \(4⋅{}^{5}\!\log(6x+1)=y-32\) \({}^{5}\!\log(6x+1)=\frac{1}{4}y-8\) 1p ○ \(6x+1=5^{\frac{1}{4}y-8}\) 1p ○ \(6x=5^{\frac{1}{4}y-8}-1\) \(x=\frac{1}{6}⋅5^{\frac{1}{4}y-8}-\frac{1}{6}\) 1p
havo wiskunde B	9.3 Rekenregels voor logaritmen
Logaritmische formules herleiden (4)	opgave 1 Herleid tot de gevraagde vorm. 3p a Schrijf de formule \(y=2{,}72⋅{}^{2}\!\log(x)+2{,}09\) in de vorm \(y={}^{2}\!\log(ax^b)\text{.}\) Geef \(a\) en \(b\) in twee decimalen. Herleiden (4) 00l0 - Logaritmische formules herleiden - basis - 0ms - dynamic variables a \(y=2{,}72⋅{}^{2}\!\log(x)+2{,}09\) \(\text{ }={}^{2}\!\log(x^{2{,}72})+2{,}09\) 1p ○ \(\text{ }={}^{2}\!\log(x^{2{,}72})+{}^{2}\!\log(2^{2{,}09})\) \(\text{ }={}^{2}\!\log(x^{2{,}72}⋅2^{2{,}09})\) 1p ○ \(\text{ }={}^{2}\!\log(x^{2{,}72}⋅4{,}257...)\) Dus \(y={}^{2}\!\log(4{,}26⋅x^{2{,}72})\text{.}\) 1p 3p b Schrijf de formule \(y={}^{5}\!\log(56x^4\sqrt{x})\) in de vorm \(y=a+b⋅{}^{5}\!\log(x)\text{.}\) Geef \(a\) in twee decimalen. Logaritmisch (5) 00l1 - Logaritmische formules herleiden - basis - 1ms - dynamic variables b \(y={}^{5}\!\log(56x^4\sqrt{x})\) \(\text{ }={}^{5}\!\log(56x^{4{,}5})\) 1p ○ \(\text{ }={}^{5}\!\log(56)+{}^{5}\!\log(x^{4{,}5})\) \(\text{ }={}^{5}\!\log(56)+4{,}5⋅{}^{5}\!\log(x)\) 1p ○ \(\text{ }=2{,}501...+4{,}5⋅{}^{5}\!\log(x)\) Dus \(y=2{,}50+4{,}5⋅{}^{5}\!\log(x)\text{.}\) 1p 3p c Schrijf de formule \(y={}^{4}\!\log(2{,}8x)+2{,}8\) in de vorm \(y=a+b⋅{}^{3}\!\log(x)\text{.}\) Geef \(a\) en \(b\) in twee decimalen. Herleiden (6) 00l2 - Logaritmische formules herleiden - basis - 0ms - dynamic variables c \(y={}^{4}\!\log(2{,}8x)+2{,}8\) \(\text{ }={}^{4}\!\log(2{,}8)+{}^{4}\!\log(x)+2{,}8\) 1p ○ \(\text{ }={}^{4}\!\log(2{,}8)+2{,}8+{{}^{3}\!\log(x) \over {}^{3}\!\log(4)}\) \(\text{ }={}^{4}\!\log(2{,}8)+2{,}8+{1 \over {}^{3}\!\log(4)}⋅{}^{3}\!\log(x)\) 1p ○ \(\text{ }=0{,}742...+2{,}8+{1 \over 1{,}261...}⋅{}^{3}\!\log(x)\) \(\text{ }=3{,}542...+0{,}792...⋅{}^{3}\!\log(x)\) Dus \(y=3{,}54+0{,}79⋅{}^{3}\!\log(x)\text{.}\) 1p 3p d Schrijf de formule \(y=8⋅{}^{3}\!\log(36x)+10\) in de vorm \(y=a+b⋅{}^{3}\!\log(4x)\text{.}\) Herleiden (7) 00l3 - Logaritmische formules herleiden - basis - 1ms - dynamic variables d \(y=8⋅{}^{3}\!\log(36x)+10\) \(\text{ }=8⋅({}^{3}\!\log(9)+{}^{3}\!\log(4x))+10\) 1p ○ \(\text{ }=8⋅(2+{}^{3}\!\log(4x))+10\) 1p ○ \(\text{ }=16+8⋅{}^{3}\!\log(4x)+10\) \(\text{ }=26+8⋅{}^{3}\!\log(4x)\) 1p
havo wiskunde B	9.4 Formules omwerken
Logaritmische formules herleiden (6)	opgave 1 Herleid tot de gevraagde vorm. 3p a Schrijf de formule \(y=7\,100⋅1{,}28^x\) 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 \(y=7\,100⋅1{,}28^x\) \(\log(y)=\log(7\,100⋅1{,}28^x)\) \(\log(y)=\log(7\,100)+\log(1{,}28^x)\) 1p ○ \(\log(y)=\log(7\,100)+x⋅\log(1{,}28)\) 1p ○ \(\log(y)=3{,}851...+x⋅0{,}10720...\) Dus \(\log(y)=0{,}1072x+3{,}85\) 1p 3p b Schrijf de formule \(y=3\,000⋅0{,}79^{6x+5}\) 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 \(y=3\,000⋅0{,}79^{6x+5}\) \(\log(y)=\log(3\,000⋅0{,}79^{6x+5})\) \(\log(y)=\log(3\,000)+\log(0{,}79^{6x+5})\) 1p ○ \(\log(y)=\log(3\,000)+(6x+5)⋅\log(0{,}79)\) \(\log(y)=\log(3\,000)+6x⋅\log(0{,}79)+5⋅\log(0{,}79)\) 1p ○ \(\log(y)=3{,}477...+6x⋅-0{,}10237...+5⋅-0{,}10237...\) \(\log(y)=3{,}477...-0{,}61423...⋅x-0{,}51186...\) Dus \(\log(y)=-0{,}6142x+2{,}97\) 1p 3p c Schrijf de formule \(\log(y)=0{,}5462x+1{,}42\) 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 \(\log(y)=0{,}5462x+1{,}42\) \(y=10^{0{,}5462x+1{,}42}\) 1p ○ \(y=10^{0{,}5462x}⋅10^{1{,}42}\) \(y=(10^{0{,}5462})^x⋅10^{1{,}42}\) 1p ○ \(y=3{,}517...^x⋅26{,}302...\) Dus \(y=26⋅3{,}52^x\text{.}\) 1p 3p d Schrijf de formule \(\log(y)=2{,}52-1{,}76⋅\log(x)\) in de vorm \(y=ax^b\text{.}\) Geef \(a\) in gehelen. Dubbel (3) 00kr - Logaritmische formules herleiden - basis - 0ms - dynamic variables d \(\log(y)=2{,}52-1{,}76⋅\log(x)\) \(\log(y)=\log(10^{2{,}52})+\log(x^{-1{,}76})\) \(\log(y)=\log(10^{2{,}52}⋅x^{-1{,}76})\) 1p ○ \(y=10^{2{,}52}⋅x^{-1{,}76}\) 1p ○ \(y=331{,}131...⋅x^{-1{,}76}\) Dus \(y=331⋅x^{-1{,}76}\text{.}\) 1p opgave 2 Herleid tot de gevraagde vorm. 3p a Schrijf de formule \(y=330x^{-1{,}26}\) in de vorm \(\log(y)=a+b⋅\log(x)\text{.}\) Geef \(a\) in twee decimalen. Dubbel (1) 00ks - Logaritmische formules herleiden - basis - 0ms - dynamic variables a \(y=330x^{-1{,}26}\) \(\log(y)=\log(330x^{-1{,}26})\) 1p ○ \(\log(y)=\log(330)+\log(x^{-1{,}26})\) \(\log(y)=\log(330)-1{,}26⋅\log(x)\) 1p ○ \(\log(y)=2{,}518...-1{,}26⋅\log(x)\) Dus \(y=2{,}52-1{,}26⋅\log(x)\text{.}\) 1p 3p b Schrijf de formule \(y={510 \over x^3\sqrt{x}}\) in de vorm \(\log(y)=a+b⋅\log(x)\text{.}\) Geef \(a\) in twee decimalen. Dubbel (2) 00kt - Logaritmische formules herleiden - basis - 0ms - dynamic variables b \(y={510 \over x^3\sqrt{x}}=510x^{-3{,}5}\) \(\log(y)=\log(510x^{-3{,}5})\) 1p ○ \(\log(y)=\log(510)+\log(x^{-3{,}5})\) \(\log(y)=\log(510)-3{,}5⋅\log(x)\) 1p ○ \(\log(y)=2{,}707...-3{,}5⋅\log(x)\) Dus \(y=2{,}71-3{,}5⋅\log(x)\text{.}\) 1p

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