need to find the exp button for population increase of 3% in 10 years based on initial population of 100000.

it does not matter what the original population is.

$$\\1.03=1(1+r)^{10}\\\\
1.03=(1+r)^{10}\\\\
log(1.03)=log(1+r)^{10}\\\\
log(1.03)=10log(1+r)}\\\\
10^{\left(\frac{log(1.03)}{10}\right)}=10^{log(1+r)}}\\\\
10^{\left(\frac{log(1.03)}{10}\right)}=1+r\\\\
r=10^{\left(\frac{log(1.03)}{10}\right)}-1\\\\$$

 

$${{\mathtt{10}}}^{\left({\frac{{log}_{10}\left({\mathtt{1.03}}\right)}{{\mathtt{10}}}}\right)}{\mathtt{\,-\,}}{\mathtt{1}} = {\mathtt{0.002\: \!960\: \!253\: \!145\: \!653\: \!4}}$$

 

so the yearly rate of increase is approximately    0.296%