Question

What is the solution to 2log_9(x)=log_9(8)+log_9(x-2)
x=-4
x=-2
x=4
x=8

Answer 1

x = 4

Explanation:

`Domain:\\\\x>0\ \wedge\ x-2>0\\\\x>0\ \wedge\ x>2\\\\\boxed{D:\ x>2}\\\\=============================`

`2\log_9x=\log_98+\log_9(x-2)\\\\\text{use}\ n\log_ab=\log_ab^n\ \text{and}\ \log_ab+\log_ac=\log_a(bc)\\\\\log_9x^2=\log_9\bigg(8(x-2)\bigg)\iff x^2=8(x-2)\qquad\text{use the distributive property}\\\\x^2=8x-16\qquad\text{subtract 8x from both sides}\\\\x^2-8x=-16\qquad\text{add 16 to both sides}\\\\x^2-8x+16=0\\\\x^2-4x-4x+16=0\\\\x(x-4)-4(x-4)=0\\\\(x-4)(x-4)=0\\\\(x-4)^2=0\iff x-4=0\qquad\text{add 4 to both sides}\\\\x=4\in D`