Did I do solve the following equation properly using the laws of logs + exponentials?

Solve:

$\displaystyle log_5(log_4(log_3x)) = 1$

My Work:

$\displaystyle log_5(log_4(log_3x)) = 5^{log_5(log_4(log_3x))} = log_4(log_3x) = 5^{1} = 5$

$\displaystyle (log_4(log_3x) = 4^{log_4(log_3x)} = 4^{5} = 1024$

$\displaystyle log_3x = 3^{log_3x} = x = 3^{1024}$

Answer: $\displaystyle 3^{1024}$