Question 238822
[6log(2)3] / [1 + log(2)3] = [6ln3] / [ln2 + ln3]
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Working with the left side:
6[ln(3)/ln(2)] / [1 + ln(3)/ln(2)]
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= [6ln(3)]/[ln(2)] divided by [ln(2) + ln(3)]/[ln(2)]
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If you invert the denominator and multiply the ln(2)'s cancel 
and you end up with:
= [6ln(3)] divided by [ln(2) + ln(3)]
which is the right side you your original equation.
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Comment: The key here is that you see that log(2)3 = ln(3)/ln(2)
That is using the Change of base Rule.
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Cheers,
Stan H.