Exercise Zone : Persamaan Logaritma

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

Tipe :

Standar SBMPTN HOTS

No. 1

\log_4\left(\log_2x\right)+\log_2\left(\log_4x\right)=2

SUMBER

\(\begin{aligned} \log_4\left(\log_2x\right)+\log_2\left(\log_4x\right)&=2\\ \log_4\left(\log_2x\right)+\log_{2^2}\left(\log_{2^2}x\right)^2&=\log_44^2\\ \log_4\left(\log_2x\right)+\log_4\left(\dfrac12\log_2x\right)^2&=\log_416\\ \log_4\left(\log_2x\right)+\log_4\left(\dfrac14\left(\log_2x\right)^2\right)&=\log_416\\ \log_4\left(\log_2x\cdot\dfrac14\left(\log_2x\right)^2\right)&=\log_416\\ \dfrac14\left(\log_2x\right)^3&=16\\ \left(\log_2x\right)^3&=64\\ \log_2x&=4\\ x&=2^4\\ &=\boxed{\boxed{16}} \end{aligned}\)

Jadi, x=16.

No. 2

Tentukan nilai x jika {{^2\negmedspace\log(3x-2)}+{^2\negmedspace\log9}-{^2\negmedspace\log x}=3}

Grup Telegram

\(\eqalign{ {^2\negmedspace\log(3x-2)}+{^2\negmedspace\log9}-{^2\negmedspace\log x}&=3\\ {^2\negmedspace\log\dfrac{(3x-2)9}x}&={^2\negmedspace\log2^3}\\ {^2\negmedspace\log\dfrac{27x-18}x}&={^2\negmedspace\log8}\\ \dfrac{27x-18}x&=8\\ 27x-18&=8x\\ 19x&=18\\ x&=\boxed{\boxed{\dfrac{18}{19}}} }\)

Jadi, x=\dfrac{18}{19}.

No. 3

Nilai x yang memenuhi persamaan {{^{10}\negmedspace\log(2x-5)}={^{10}\negmedspace\log(x+3)}}

Grup Telegram

\(\eqalign{ {^{10}\negmedspace\log(2x-5)}&={^{10}\negmedspace\log(x+3)}\\ 2x-5&=x+3\\ x&=\boxed{\boxed{8}} }\)

Jadi, x=8.

No. 4

HP dari persamaan logaritma {{^2\negmedspace\log(2x+3)}-{^2\negmedspace\log(3x-4)}=3} adalah...

Grup Telegram

\(\eqalign{ {^2\negmedspace\log(2x+3)}-{^2\negmedspace\log(3x-4)}&=3\\ {^2\negmedspace\log\dfrac{2x+3}{3x-4}}&=3\\ \dfrac{2x+3}{3x-4}&=2^3\\ \dfrac{2x+3}{3x-4}&=8\\ \dfrac{2x+3}{3x-4}-8&=0\\ \dfrac{2x+3-8(3x-4)}{3x-4}&=0\\ \dfrac{2x+3-24x+32}{3x-4}&=0\\ \dfrac{-22x+35}{3x-4}&=0\\ -22x+35&=0\\ -22x&=-35\\ x&=\dfrac{-35}{-22}\\ &=\boxed{\boxed{\dfrac{35}{22}}} }\)

Jadi, Hp = \left\{\dfrac{35}{22}\right\}.

No. 4

Tentukan nilai x jika {{^3\negmedspace\log x}