Logarithmic Expression Addition and Subtraction with x Variables

Question

I'm trying to solve the following log problem:

Below are my steps:

But my steps appear wrong as the solution is 6. What am I doing wrong?

Answer 1

$(900 x^2)^{\frac 12} \ne 900(x^2)^{\frac 12}$ was your mistake.

$\log_{\sqrt3}(18x^2) - \log_{\sqrt 3}(20x^3) + \frac 12 \log_{\sqrt 3}(900 x^2)=$

$\log_{\sqrt 3}\frac {18x^2}{20x^3} + \log_{\sqrt 3} (900x^2)^{\frac 12} =$

$\log_{\sqrt 3}{\frac 9{10x}} + \log_{\sqrt 3}(30x)=$

$\log_{\sqrt 3}(\frac 9{10x}\cdot 30x)=\log_{\sqrt 3}27=$

$\log_{\sqrt 3} 3^3 =\log_{\sqrt 3} \sqrt 3^6 = 6$.