How can i calculate first digit in $(13)^{139}$. How can i calculate first digit in  $(13)^{139}$.

$\bf{My\; Try}::$ Let $a$ be the first Digit of $13^{139}$.

So $10^{b}a &lt; 13^{139}&lt;10^{b}(a+1)$, where $b= \lfloor \log_{10}(13^{139})\rfloor$

Now taking $\log_{10}$ on both side

$b+\log_{10}(a)&lt;139\cdot \log_{10}(13)&lt;b+\log_{10}(a+1)$

Now I did not understand how can i solve after that 

Help Required

Thanks
Take your favourite calculator, and calculate $139 \log_{10}13$. Now let $x$ be the fractional part of this, and calculate $10^x$. Deduce the result.
$13$ is congruent to $3  (\mod 10)$, so $13^4$ is congruent to $3^4$ or $81$ congruent to $1 (\mod 10)$, but $139$ is the product of 34 and 4 added to 3... So $13^{136}$ is congruent to $1 (\mod 10)$ so $13^{139}$ is congruent to $13^3$ or $7 (\mod 10)$ so the first digit is $7$.