Pembahasan soal logaritma (2)


Soal nomor 4.
jika a^{2}=b^{3}=c^{5}=d^{6}, tunjukkan bahwa {}^{d}\log(abc)=\frac{31}{5}.

Jawab.

Diketahui a^{2}=b^{3}=c^{5}=d^{6}, diperoleh
a^{2}=d^{6},sehingga \log(a)^{2}=\log(d)^{6}
2.\log(a)=6.\log(d), dan diperoleh \frac{\log(a)}{\log(d)}=3
{}^{d}\log(a)=3 ...(1). analog dengan cara ini dari b^{3}=d^{6} diperoleh {}^{d}\log(b)=2 ...(2)
c^{5}=d^{6} diperoleh {}^{d}\log(c)=\frac{6}{5} ...(3).
persamaan (1)+(2)+(3) diperoleh
{}^{d}\log(a)+{}^{d}\log(b)+{}^{d}\log(c)=3+2+\frac{6}{5}
{}^{d}\log(a)+{}^{d}\log(b)+{}^{d}\log(c)=\frac{31}{5}
{}^{d}\log(a.b.c)=\frac{31}{5}

Soal nomor 5.

Jika \frac{\log(x)}{b-c}=\frac{\log(y)}{c-a}=\frac{\log(z)}{a-b}, tunjukkan bahwa x.y.z=1

Jawab.

dari \frac{\log(x)}{b-c}=\frac{\log(y)}{c-a}=\frac{\log(z)}{a-b},
\frac{\log(x)}{b-c}=\frac{\log(y)}{c-a}
(c-a)\log(x)=(b-c)\log(y)
\frac{\log(y)}{\log(x)}=\frac{(c-a)}{(b-c)}
{}^{x}\log(y)=\frac{(c-a)}{(b-c)}
y =x^{\frac{c-a}{b-c}} …. (1)
dan dari \frac{\log(x)}{b-c}=\frac{\log(z)}{a-b}
\frac{{a-b}}{b-c}=\frac{\log(z)}{\log(x)}
\frac{{a-b}}{b-c}={}^{x}\log(z), sehingga
z=x^{\frac{{a-b}}{b-c}} …. (2)
sehingga x.y.z = x.x^{\frac{c-a}{b-c}}.x^{\frac{a-b}{b-c}}
x.y.z = x^{1+\frac{c-a}{b-c}+\frac{a-b}{b-c}}

x.y.z = x ^{\frac{b-c}{b-c}+\frac{c-a}{b-c}+\frac{a-b}{b-c}}

x.y.z = x^{\frac{b-c+c-a+a-b}{b-c}}
x.y.z = x^{0} = 1

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