Taming the Radix

The length of the numeral $19^{1234}$ will be 1 more than the highest power of 3 which divides $19^{1234}$ . We can formulate this into logarithms and then simplify the answer.

$=\left\lfloor \log _{ 3 }{ { 19 }^{ 1234 } } \right\rfloor +1\\ =\left\lfloor 1234\log _{ 3 }{ 19 } \right\rfloor +1$

Since $\log _{ 3 }{ 19 } \approx 2.68$ ,

$=3307+1\\ =3308$