Converting $29^{1312000}t$ o base $10$

Question

I am trying to do some calculations with the number $29^{1312000}$ and I find it would be much easier if I could convert it (approximately) to a base $10$ number. The closest I could come was to start with $30$, and note that every other power added a power of ten over what you would have with just base $10$ - so a rough approximation would be $10^{1968000}$. I'd be interested in having a better estimate.

Answer 1

$$29^{1312000}=10^{\log_{10}29^{1312000}}=10^{1312000\log_{10}29}\approx10^{1312000*1.46239}\approx10^{1918666.17324}$$

Answer 2

Hi.
Recall $a^{log_a(b)}=b$, so $10^{log_{10}(29^{1312000})}=29^{1312000}$.
A better estimate would be $10^{1918666}$.