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

$\begin{array}{rl}3{}^x\log \left(3y\right)& =k\\ {}^x\log \left(3y\right)& ={\displaystyle \frac{k}{3}}\\ 3y& =x^{\frac{k}{3}}\end{array}$

$\begin{array}{rl}3({}^{3x}\log (27z))& =k\\ {}^{3x}\log (27z)& ={\displaystyle \frac{k}{3}}\\ 27z& =(3x{)}^{\frac{k}{3}}\\ & =3^{\frac{k}{3}}{x}^{\frac{k}{3}}\end{array}$

$\begin{array}{rl}{}^{3x^4}\log (81yz)& =k\\ 81yz& ={\left(3x^4\right)}^k\\ 3y\cdot 27z& =3^k{x}^{4k}\\ x^{\frac{k}{3}}\cdot 3^{\frac{k}{3}}{x}^{\frac{k}{3}}& =3^k{x}^{4k}\\ 3^{\frac{k}{3}}{x}^{\frac{2k}{3}}& =3^k{x}^{4k}\\ {\left(3^{\frac{k}{3}}{x}^{\frac{2k}{3}}\right)}^{\frac{3}{k}}& ={\left(3^k{x}^{4k}\right)}^{\frac{3}{k}}\\ 3x^2& =3^3{x}^{12}\\ x^{10}& =3^{-2}\\ x& =3^{-\frac{1}{5}}\end{array}$

$\begin{array}{rl}3{}^x\log \left(3y\right)& ={}^{3x^4}\log (81yz)\\ {}^{3^{-\frac{1}{5}}}\log {\left(3y\right)}^3& ={}^{3{\left(3^{-\frac{1}{5}}\right)}^4}\log \left(3^{4}yz\right)\\ -5{}^3\log \left(3^3{y}^3\right)& ={}^{3\left(3^{-\frac{4}{5}}\right)}\log \left(3^{4}yz\right)\\ -5{}^3\log \left(3^3{y}^3\right)& ={}^{3^{\frac{1}{5}}}\log \left(3^{4}yz\right)\\ -5{}^3\log \left(3^3{y}^3\right)& =5{}^3\log \left(3^{4}yz\right)\\ -{}^3\log \left(3^3{y}^3\right)& ={}^3\log \left(3^{4}yz\right)\\ 3^{-3}{y}^{-3}& =3^{4}yz\\ 3^{-7}& =y^{4}z\end{array}$

$\begin{array}{rl}{x}^5{y}^{4}z& ={\left(3^{-\frac{1}{5}}\right)}^5{3}^{-7}\\ & =3^{-1}{3}^{-7}\\ & =3^{-8}\\ & ={\displaystyle \frac{1}{3^8}}\\ & =\boxed{{\displaystyle \boxed{{\displaystyle {\displaystyle \frac{1}{6561}}}}}}\end{array}$

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