Prove that $y^2 + ky + (k-2) = 0$

Question

I am given that,

$y=\log_{3}{x}$ and

$\log_{3}{x} - \log_{x}{9}+\log{3^k} + k\log_{x}{3}=0$

I am asked to prove that

$\boxed{y^2 + ky + (k-2) = 0}$

I assumed that $\log{3^k} = \log_{10}{3^k}$

After many tries of failing I got an equation which I should find the value of $\log_{10}{3}$ in order to continue forward but the problem is I am not allowed to use the calculator or the log table in my country

Could any folks over here could help me with this problem or if the question itself is wrong could any of yall mention it

Thank you in advance

Answer 1

Instead of $\log3^k=\log_{10}3^k,$ assume $\log3^k$ meant $\log_33^k.$

Then, the hypothesis rewrites $$y-\frac2y+k+\frac ky=0$$ and the result follows.