Question

solve exactly for a in the equation:

log a^5 + log a^(3/2) = log 625
4 2 8

the 4,2,8 are the bases

Answer 1

log4 a^5 + log2 a^(3/2) = log8 625

Answer 2

let's convert to the same base 2

log₄ a^5 = log₂ a^5 /log₂ 4
= (1/2)(log₂ a^5) since log₂ 4 = 2

similary log₈ 625
= log₂ 625/log₂ 8

so the new equation is
(1/2)(log₂ a^5) + log₂ a^(3/2) = (1/3)log₂625

using the rules of logs

log₂ a^(5/2) + log₂ a^(3/2) = log₂ 625^(1/3)

log₂ a^4 = log₂ 625^(1/3)

a^4 = 625^(1/3)
a = 5^(1/3) or the cuberoot of 5