**5.5 Indeks dan Logaritma, SPM Praktis (Soalan Pendek)**

**Soalan 9**

Selesaikan persamaan, ${\mathrm{log}}_{2}4x=1-{\mathrm{log}}_{4}x$

*Penyelesaian:*

$\begin{array}{l}{\mathrm{log}}_{2}4x=1-{\mathrm{log}}_{4}x\\ {\mathrm{log}}_{2}4x=1-\frac{{\mathrm{log}}_{2}x}{{\mathrm{log}}_{2}4}\\ {\mathrm{log}}_{2}4x=1-\frac{{\mathrm{log}}_{2}x}{2}\\ 2{\mathrm{log}}_{2}4x=2-{\mathrm{log}}_{2}x\\ {\mathrm{log}}_{2}16{x}^{2}={\mathrm{log}}_{2}4-{\mathrm{log}}_{2}x\\ {\mathrm{log}}_{2}16{x}^{2}={\mathrm{log}}_{2}\frac{4}{x}\\ 16{x}^{2}=\frac{4}{x}\\ {x}^{3}=\frac{4}{16}=\frac{1}{4}\\ x={\left(\frac{1}{4}\right)}^{\frac{1}{3}}=0.62996\end{array}$
**Soalan 10**

Selesaikan persamaan, ${\mathrm{log}}_{4}x=25{\mathrm{log}}_{x}4$

*Penyelesaian:*

$\begin{array}{l}{\mathrm{log}}_{4}x=25{\mathrm{log}}_{x}4\\ \frac{1}{{\mathrm{log}}_{x}4}=25{\mathrm{log}}_{x}4\\ \frac{1}{25}={\left({\mathrm{log}}_{x}4\right)}^{2}\\ {\mathrm{log}}_{x}4=\pm \frac{1}{5}\\ {\mathrm{log}}_{x}4=\frac{1}{5}\text{or}{\mathrm{log}}_{x}4=-\frac{1}{5}\\ 4={x}^{\frac{1}{5}}\text{}4={x}^{-\frac{1}{5}}\\ x={4}^{5}\text{}4=\frac{1}{{x}^{\frac{1}{5}}}\\ x=1024\text{}{x}^{\frac{1}{5}}=\frac{1}{4}\\ \text{}x=\frac{1}{1024}\end{array}$
**Soalan 11**

Selesaikan persamaan, $2{\mathrm{log}}_{x}5+{\mathrm{log}}_{5}x=\mathrm{lg}1000$

*Penyelesaian:*

$\begin{array}{l}2{\mathrm{log}}_{x}5+{\mathrm{log}}_{5}x=\mathrm{lg}1000\\ 2.\frac{1}{{\mathrm{log}}_{5}x}+{\mathrm{log}}_{5}x=3\\ \times \left({\mathrm{log}}_{5}x\right)\to \text{}2+{\left({\mathrm{log}}_{5}x\right)}^{2}=3{\mathrm{log}}_{5}x\\ {\left({\mathrm{log}}_{5}x\right)}^{2}-3{\mathrm{log}}_{5}x+2=0\\ \left({\mathrm{log}}_{5}x-2\right)\left({\mathrm{log}}_{5}x-1\right)=0\\ {\mathrm{log}}_{5}x=2\text{or}{\mathrm{log}}_{5}x=1\\ x={5}^{2}\text{}x=5\\ x=25\end{array}$