2074-2006=68 years. If the growth rate is 7% per year then the growth factor is 1.07^68=99.56275 approx.
Multiply by 100,000,000=9,956,275,000.
(1+r)^2=1+2r+r^2=1+2r approx for small r. This is the doubling formula. 68=64+4=2^6+2^2.
If we apply the formula 6 times we get an approximate value for 1.07^64 (r=0.07):
1.07^2=1+2*0.07=1+0.14=1.14 (true value: 1.1449)
Now put r=0.14
1.07^4=1.14^2=1.28 (1.3107...)
1.07^8=1.28^2=1.56 (1.7181...)
1.07^16=1.56^2=1+1.12=2.12 (2.9521...)
2.12=2*1.06=2(1+0.06); 1.07^32=2^2*1.06^2=4*1.12=4.48 (8.7152...)
4.48=4*1.12; 1.07^64=4^2(1.12)^2=16*1.24=16+3.84=19.84 (75.9559...)
1.07^68=1.07^64 * 1.07^4=19.84*1.28=25.3952 (99.56274...)
So, using the doubling formula we have an underestimate of 100,000,000*25.4=2,540,000,000 instead of about 10,000,000,000.
The doubling time formula should give a better approximation: when the growth is double (1+r)^T=2 where T is the time for growth to be doubled, so T=log(2)/log(1+0.07)=ln(2)/0.07 approx.=9.9 years. Call this 10 years.
In 68 years then, the population should have doubled about 6.8 times. Let's work out doubling 6 times: 2^6=64 so after 6*10=60 years (2066) the population will be 6,400,000,000. Now, using our approximate doubling formula from earlier we can estimate for the remaining 8 years. We simply multiply by 1.56 (1.07^8) to give 9,984,000,000, which is very close to the true value of 1.07^68 * 100,000,000.
If we use simple proportion to find the growth over the final 8 years, we would use the factor 8/10 * 2=1.6. This produces 1.6*64,000,000,000=102,400,000,000, which is an overestimate.