Solve the following equations forĀ x:
(i) $7^{2 x+3}=1$
(ii) $2^{x+1}=4^{x-3}$
(iii) $2^{5 x+3}=8^{x+3}$
(iv) $4^{2 x}=\frac{1}{32}$
(v) $4^{x-1} \times(0.5)^{3-2 x}=\left(\frac{1}{8}\right)^{x}$
(vi) $2^{3 x-7}=256$
(i)
$7^{2 x+3}=1$
$\Rightarrow 7^{2 x+3}=7^{0}$
$\Rightarrow 2 x+3=0$
$\Rightarrow 2 x=-3$
$\Rightarrow x=-\frac{3}{2}$
(ii)
$2^{x+1}=4^{x-3}$
$\Rightarrow 2^{x+1}=\left(2^{2}\right)^{x-3}$
$\Rightarrow 2^{x+1}=\left(2^{2 x-6}\right)$
$\Rightarrow x+1=2 x-6$
$\Rightarrow x=7$
(iii)
$2^{5 x+3}=8^{x+3}$
$\Rightarrow 2^{5 x+3}=\left(2^{3}\right)^{x+3}$
$\Rightarrow 2^{5 x+3}=2^{3 x+9}$
$\Rightarrow 5 x+3=3 x+9$
$\Rightarrow 2 x=6$
$\Rightarrow x=3$
(iv)
$4^{2 x}=\frac{1}{32}$
$\Rightarrow\left(2^{2}\right)^{2 x}=\frac{1}{2^{5}}$
$\Rightarrow 2^{4 x} \times 2^{5}=1$
$\Rightarrow 2^{4 x+5}=2^{0}$
$\Rightarrow 4 x+5=0$
$\Rightarrow x=-\frac{5}{4}$
(v)
$4^{x-1} \times(0.5)^{3-2 x}=\left(\frac{1}{8}\right)^{x}$
$\Rightarrow\left(2^{2}\right)^{x-1} \times\left(\frac{1}{2}\right)^{3-2 x}=\left(\frac{1}{2^{3}}\right)^{x}$
$\Rightarrow\left(2^{2}\right)^{x-1} \times\left(2^{-1}\right)^{3-2 x}=\left(2^{-3}\right)^{x}$
$\Rightarrow 2^{2 x-2} \times 2^{2 x-3}=2^{-3 x}$
$\Rightarrow 2^{2 x-2+2 x-3}=2^{-3 x}$
$\Rightarrow 2^{4 x-5}=2^{-3 x}$
$\Rightarrow 4 x-5=-3 x$
$\Rightarrow 7 x=5$
$\Rightarrow x=\frac{5}{7}$
(vi)
$2^{3 x-7}=256$
$\Rightarrow 2^{3 x-7}=2^{8}$
$\Rightarrow 3 x-7=8$
$\Rightarrow 3 x=15$
$\Rightarrow x=5$
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