Increasing a quantity by 100% is the same as multiplying that quantity by 2 (adding the whole amount on top of what's already there). So we can say that every 5 months the established organization doubles its membership.
To see how we can then get an expression for the members of the organization after some number of months, it can be helpful to consider numbers first.
After 10 months, the old organization will have doubled its membership twice (10/5 times). So it would have 4096*2*2 members. Or 4096 * 2^2 times.
After 20 months, the old organization will have doubled its membership four times (or 20/5 times). So it would have 4096*2*2*2*2 members. Or 4096*2^4 members.
After M months, the old organization will have doubled its membership M/5 times. So it would have 4096*2^(M/5) members.
Similarly, Increasing a quantity by 700% is the same as multiplying that quantity by 8. So we can say that every 10 months the new organization multiplies it's membership by 8.
Algebraically, we can say that after M months, the new organization will have 4*8^(M/10) members.
We can now set theses quantities equal to get an equation:
4096*2^(M/5) = 4*8^(M/10)
To solve equations in which the variable is in an exponent, we can make the bases the same. Luckly, all our bases are powers of 2. So replacing 4096 with 2^12, and 4 with 2^2 and 8 with 2^3 we get:
(2^12)*2^(M/5) = (2^2)*(2^3)^(M/10)
Using the exponent rules for powers (when a quantity is raised to an exponent, and that whole quantity is raised to another exponent, we can multiply the exponents), we get:
(2^12)*2^(M/5) = (2^2)*2^(3*M/10)
Using the exponent rule for multiplication (when two terms with the same base are mutiplued, add the exponents), we get:
2^(12 + M/5) = 2^(2 + 3M/10)
So we've got 2 raised to on thing equal to two raised to another thing. That means the two exponents must be the same:
12 + M/5 = 2 + 3M/10 ---> multiplying by 10
120 + 2M = 20 + 3M
100 = M
