Answer:
[tex]\large\boxed{\large\boxed{x=4}}[/tex]
Step-by-step explanation:
The equation to solve is:
[tex]2\log_9 x=\log_9 8+\log_9 (x-2)[/tex]
1. On the left-hand side use: "The product of a constant by a logarithm is equal to the logarithm raised to the constant"
Thus, the left-hand side is:
[tex]2\log_9 x=\log_9 x^2[/tex]
2. On the right-hand side use "The sum of two logarithms with the same base is the logarithm of the product":
Then, on the right-hand side:
[tex]\log_9 8+\log_9 (x-2)=\log_9 8(x-2)[/tex]
3. Make them equal:
[tex]\log_9 x^2=\log_9 8(x-2)[/tex]
4. Since the two functions are the same, make the arguments equal:
[tex]x^2=8(x-2)[/tex]
5. Solve the equation:
[tex]x^2=8x-16\\\\x^2-8x+16=0\\\\(x-4)^2=0\\\\x-4=0\\\\x=4\leftarrow solution[/tex]
