math

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

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

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  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

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