Question

a) log2 [2+log2(x-1)]=1
b) (log4 x)2 - 4 log4 x + 3 =0
c) log2 [log3( +21)] =2
d) log4 [15+log2(x-3)] =2
​

Answer 1

a)

$\log_2{\left(2+\log_2{(x-1)}\right)}=1\iff 2+\log_2{(x-1)}=2^1$

$\Longrightarrow \log_2{(x-1)}=0\iff x-1=2^0=1\ \therefore\ \boxed{x=2}$

b)

$\left(\log_4{x}\right)^2-4\log_4{x}+3=0$

$\log_4{x}=y\iff x=4^y$

$y^2-4y+3=0\Longrightarrow a=1;\ b=-4;\ c=3$

$y=\dfrac{-b\pm\sqrt{b^2-4ac}}{2a}\Longrightarrow y=\dfrac{-(-4)\pm\sqrt{(-4)^2-4(1)(3)}}{2(1)}$

$\Longrightarrow y=\dfrac{4\pm2}{2}=2\pm1\Longrightarrow y=1\ \text{ou}\ y=3$

$y=1\Longrightarrow x=4^1\ \therefore\ \boxed{x=4}$

$y=3\Longrightarrow x=4^3\ \therefore\ \boxed{x=64}$

$\boxed{x=\left\{4,64\right\}}$

c)

$\log_2{(\log_3{(x+21)})}=2\iff \log_3{(x+21)}=2^2=4$

$\log_3{(x+21)}=4\iff x+21=3^4=81\ \therefore\ \boxed{x=60}$

d)

$\log_4{(15+\log_2{(x-3)})}=2\iff 15+\log_2{(x-3)}=4^2=16$

$\log_2{(x-3)}=1\iff x-3=2^1=2\ \therefore\ \boxed{x=5}$