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I don't know the detailed process of seeking mathematical log, and I hope that good people will explain it patiently.
Bottom changing formula:

㏒6(√24)

= 1/2[㏒6(4×6]

= 1/2[ 1+㏒6(4)]

= 1/2[ 1+2㏒6(2)]

= 1/2+㏒3(2)÷[ 1+㏒3(2)]

= 1/2+a/( 1+a)

=[ 1+a+2a]/2( 1+a)

=(3a+ 1)/2( 1+a)。

Choose a.

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