Math Olympiad Problem | Germany | How to solve!!!

[@doyouknoworjustbelieve6694]

8 = 2^3
So the equation becomes
2^(6a) = 200
6a. Log2 (2) = log2(2 x 2 x 2 x25)
6a = 3+ log2(25)
6a = 3+ 2 x log2(5)
a = (1/2) + (1/3)log2(5)

[@sas9170]

1

[@MrLinusK]

Found a lot simpler solution...
8^a*8^a = 200
log2(8^a*8^a) = log2(200)
log2(8^a) + log2(8^a) = log2(200)
2*a*log2(8) = log2(200)
6a=log2(200)
a=log2(200)/6

[@quokka_yt]

I did it like so:
8ᵃ ⋅ 8ᵃ= 200
(8ᵃ)² = 200
8²ᵃ = 200
log₈ (8²ᵃ) = log₈ 200
2a = log₈ (8 ⋅ 25)
2a = log₈ 8 + log₈ 25
2a = 1 + log₈ 25
a = (1 + log₈ 25) / 2 ≈ 1,274

[@wolemai]

Big problem here! At time 7.20 you said that a log b = log a ^b and you based the following calculations on this. This is NOT correct!
a log b = log b^a .