HOTS Zone : Persamaan Logaritma

Berikut ini adalah kumpulan soal mengenai persamaan logaritma tipe HOTS. Jika ingin bertanya soal, silahkan gabung ke grup Facebook atau Telegram.

Tipe:

Standar SBMPTN HOTS

No. 1

Misalkan x, y, z bilangan real positif yang memenuhi sistem persamaan:
3\ {^x\negmedspace\log\left(3y\right)}=3\left(^{3x}\negmedspace\log(27z)\right)={^{3x^4}\negmedspace\log(81yz)}\ne0
Nilai x^5y^4z adalah ....

SUMBER

3\ {^x\negmedspace\log\left(3y\right)}=3\left(^{3x}\negmedspace\log(27z)\right)={^{3x^4}\negmedspace\log(81yz)}=k

\(\eqalign{ 3\ {^x\negmedspace\log\left(3y\right)}&=k\\ {^x\negmedspace\log\left(3y\right)}&=\dfrac{k}3\\ 3y&=x^{\frac{k}3} }\)

\(\eqalign{ 3\left(^{3x}\negmedspace\log(27z)\right)&=k\\ ^{3x}\negmedspace\log(27z)&=\dfrac{k}3\\ 27z&=(3x)^{\frac{k}3}\\ &=3^{\frac{k}3}x^{\frac{k}3} }\)

\(\eqalign{ {^{3x^4}\negmedspace\log(81yz)}&=k\\ 81yz&=\left(3x^4\right)^k\\ 3y\cdot27z&=3^kx^{4k}\\ x^{\frac{k}3}\cdot3^{\frac{k}3}x^{\frac{k}3}&=3^kx^{4k}\\ 3^{\frac{k}3}x^{\frac{2k}3}&=3^kx^{4k}\\ \left(3^{\frac{k}3}x^{\frac{2k}3}\right)^{\frac3k}&=\left(3^kx^{4k}\right)^{\frac3k}\\ 3x^2&=3^3x^{12}\\ x^{10}&=3^{-2}\\ x&=3^{-\frac15} }\)

\(\eqalign{ 3\ {^x\negmedspace\log\left(3y\right)}&={^{3x^4}\negmedspace\log(81yz)}\\ {^{3^{-\frac15}}\negmedspace\log\left(3y\right)^3}&={^{3\left(3^{-\frac15}\right)^4}\negmedspace\log\left(3^4yz\right)}\\ -5{^3\negmedspace\log\left(3^3y^3\right)}&={^{3\left(3^{-\frac45}\right)}\negmedspace\log\left(3^4yz\right)}\\ -5{^3\negmedspace\log\left(3^3y^3\right)}&={^{3^{\frac15}}\negmedspace\log\left(3^4yz\right)}\\ -5{^3\negmedspace\log\left(3^3y^3\right)}&=5\ {^3\negmedspace\log\left(3^4yz\right)}\\ -{^3\negmedspace\log\left(3^3y^3\right)}&={^3\negmedspace\log\left(3^4yz\right)}\\ 3^{-3}y^{-3}&=3^4yz\\ 3^{-7}&=y^4z }\)

\(\eqalign{ x^5y^4z&=\left(3^{-\frac15}\right)^53^{-7}\\ &=3^{-1}3^{-7}\\ &=3^{-8}\\ &=\dfrac1{3^8}\\ &=\boxed{\boxed{\dfrac1{6561}}} }\)

Jadi, x^5y^4z=\dfrac1{6561}.