Soalan 3:
Hitung nilai x bagi setiap persamaan berikut.
$$ \text { (a) } 2^6 \div 2^x=8 $$
$$ \text { (b) } 3^{-4} \times 81=3^x $$
$$ \text { (c) } a^x a^8=1 $$
$$ \text { (d) } 4 \times 8^{x+1}=2^{2 x} $$
$$ \text { (e) }\left(a^x\right)^2 \times a^5=a^{3 x} $$
$$ \text { (f) } 2^x=\frac{2^{10}}{16^x} $$
$$ \text { (g) } 3^6 \div 3^x=81^{(x-1)} $$
$$ \text { (h) }\left(m^2\right)^x \times m^{(x+1)}=m^{-2} $$
$$ \text { (i) } 25^x \div 125=\frac{1}{5^x} $$
Penyelesaian:
(a)
$$ \begin{aligned} 2^6 \div 2^x & =8 \\ 2^{6-x} & =2^3 \\ 6-x & =3 \\ -x & =-3 \\ x & =3 \end{aligned} $$
(b)
$$ \begin{aligned} 3^{-4} \times 81 & =3^x \\ 3^{-4} \times 3^4 & =3^x \\ 3^{-4+4} & =3^x \\ 3^0 & =3^x \\ \therefore x & =0 \end{aligned} $$
(c)
$$ \begin{aligned} a^x a^8 & =1 \\ a^{x+8} & =1 \\ a^{x+8} & =a^0 \\ x+8 & =0 \\ x & =-8 \end{aligned} $$
(d)
$$ \begin{aligned} 4 \times 8^{x+1} & =2^{2 x} \\ 2^2 \times\left(2^3\right)^{(x+1)} & =2^{2 x} \\ 2^2 \times 2^{3 x+3} & =2^{2 x} \\ 2^{2+3 x+3} & =2^{2 x} \\ 2^{3 x+5} & =2^{2 x} \\ 3 x+5 & =2 x \\ 3 x-2 x & =-5 \\ x & =-5 \end{aligned} $$
(e)
$$ \begin{aligned} \left(a^x\right)^2 \times a^5 & =a^{3 x} \\ a^{2 x} \times a^5 & =a^{3 x} \\ a^{2 x+5} & =a^{3 x} \\ 2 x+5 & =3 x \\ 2 x-3 x & =-5 \\ -x & =-5 \\ x & =5 \end{aligned} $$
(f)
$$ \begin{aligned} 2^x & =\frac{2^{10}}{16^x} \\ 2^x & =\frac{2^{10}}{\left(2^4\right)^x} \\ 2^x & =2^{10-4 x} \\ x & =10-4 x \\ x+4 x & =10 \\ 5 x & =10 \\ x & =\frac{10}{5} \\ x & =2 \end{aligned} $$
(g)
$$ \begin{aligned} 3^6 \div 3^x & =81^{(x-1)} \\ 3^{6-x} & =\left(3^4\right)^{(x-1)} \\ 3^{6-x} & =3^{4 x-4} \\ 6-x & =4 x-4 \\ -x-4 x & =-4-6 \\ -5 x & =-10 \\ 5 x & =10 \\ x & =\frac{10}{5} \\ x & =2 \end{aligned} $$
(h)
$$ \begin{aligned} \left(m^2\right)^x \times m^{(x+1)} & =m^{-2} \\ m^{2 x} \times m^{(x+1)} & =m^{-2} \\ m^{2 x+x+1} & =m^{-2} \\ m^{3 x+1} & =m^{-2} \\ 3 x+1 & =-2 \\ 3 x & =-3 \\ x & =\frac{-3}{3} \\ x & =-1 \end{aligned} $$
(i)
$$ \begin{aligned} 25^x \div 125 & =\frac{1}{5^x} \\ 5^{2 x} \div 5^3 & =5^{-x} \\ 5^{2 x-3} & =5^{-x} \\ 2 x-3 & =-x \\ 2 x+x & =3 \\ 3 x & =3 \\ x & =1 \end{aligned} $$
Hitung nilai x bagi setiap persamaan berikut.
$$ \text { (a) } 2^6 \div 2^x=8 $$
$$ \text { (b) } 3^{-4} \times 81=3^x $$
$$ \text { (c) } a^x a^8=1 $$
$$ \text { (d) } 4 \times 8^{x+1}=2^{2 x} $$
$$ \text { (e) }\left(a^x\right)^2 \times a^5=a^{3 x} $$
$$ \text { (f) } 2^x=\frac{2^{10}}{16^x} $$
$$ \text { (g) } 3^6 \div 3^x=81^{(x-1)} $$
$$ \text { (h) }\left(m^2\right)^x \times m^{(x+1)}=m^{-2} $$
$$ \text { (i) } 25^x \div 125=\frac{1}{5^x} $$
Penyelesaian:
(a)
$$ \begin{aligned} 2^6 \div 2^x & =8 \\ 2^{6-x} & =2^3 \\ 6-x & =3 \\ -x & =-3 \\ x & =3 \end{aligned} $$
(b)
$$ \begin{aligned} 3^{-4} \times 81 & =3^x \\ 3^{-4} \times 3^4 & =3^x \\ 3^{-4+4} & =3^x \\ 3^0 & =3^x \\ \therefore x & =0 \end{aligned} $$
(c)
$$ \begin{aligned} a^x a^8 & =1 \\ a^{x+8} & =1 \\ a^{x+8} & =a^0 \\ x+8 & =0 \\ x & =-8 \end{aligned} $$
(d)
$$ \begin{aligned} 4 \times 8^{x+1} & =2^{2 x} \\ 2^2 \times\left(2^3\right)^{(x+1)} & =2^{2 x} \\ 2^2 \times 2^{3 x+3} & =2^{2 x} \\ 2^{2+3 x+3} & =2^{2 x} \\ 2^{3 x+5} & =2^{2 x} \\ 3 x+5 & =2 x \\ 3 x-2 x & =-5 \\ x & =-5 \end{aligned} $$
(e)
$$ \begin{aligned} \left(a^x\right)^2 \times a^5 & =a^{3 x} \\ a^{2 x} \times a^5 & =a^{3 x} \\ a^{2 x+5} & =a^{3 x} \\ 2 x+5 & =3 x \\ 2 x-3 x & =-5 \\ -x & =-5 \\ x & =5 \end{aligned} $$
(f)
$$ \begin{aligned} 2^x & =\frac{2^{10}}{16^x} \\ 2^x & =\frac{2^{10}}{\left(2^4\right)^x} \\ 2^x & =2^{10-4 x} \\ x & =10-4 x \\ x+4 x & =10 \\ 5 x & =10 \\ x & =\frac{10}{5} \\ x & =2 \end{aligned} $$
(g)
$$ \begin{aligned} 3^6 \div 3^x & =81^{(x-1)} \\ 3^{6-x} & =\left(3^4\right)^{(x-1)} \\ 3^{6-x} & =3^{4 x-4} \\ 6-x & =4 x-4 \\ -x-4 x & =-4-6 \\ -5 x & =-10 \\ 5 x & =10 \\ x & =\frac{10}{5} \\ x & =2 \end{aligned} $$
(h)
$$ \begin{aligned} \left(m^2\right)^x \times m^{(x+1)} & =m^{-2} \\ m^{2 x} \times m^{(x+1)} & =m^{-2} \\ m^{2 x+x+1} & =m^{-2} \\ m^{3 x+1} & =m^{-2} \\ 3 x+1 & =-2 \\ 3 x & =-3 \\ x & =\frac{-3}{3} \\ x & =-1 \end{aligned} $$
(i)
$$ \begin{aligned} 25^x \div 125 & =\frac{1}{5^x} \\ 5^{2 x} \div 5^3 & =5^{-x} \\ 5^{2 x-3} & =5^{-x} \\ 2 x-3 & =-x \\ 2 x+x & =3 \\ 3 x & =3 \\ x & =1 \end{aligned} $$