Question

Solve the logarthmic equations. Round answers to the nearest hundredth.

log2(x - 6) + log2(x - 4) = log2 x

x = 8, 3
x = 8
no solution

Answer 1

$\text{The domain:}\\x-6 \ \textgreater \ 0\ \wedge\ x-4 \ \textgreater \ 0\ \wedge\ x \ \textgreater \ 0\\x \ \textgreater \ 6\ \wedge\ x \ \textgreater \ 4\ \wedge\ x\ \textgreater \ 0\\\\D:x\in(6;\ \infty)\\\\\log_2(x-6)+\log_2(x-4)=\log_2x\\\\\log_2[(x-6)(x-4)]=\log_2x\iff(x-6)(x-4)=x\\\\x^2-4x-6x+24=x\\\\x^2-10x+24=x\ \ \ |-x\\\\x^2-11x+24=0\\\\x^2-8x-3x+24=0\\\\x(x-8)-3(x-8)=0\\\\(x-8)(x-3)=0\iff x-8=0\ \vee\ x-3=0\\\\x=8\in D\ \vee\ x=3\notin D\\\\\text{Answer:}\ x=8$