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9514 1404 393

  (2ac+1)/(abc+2c+1)

Step-by-step explanation:

It helps to understand the factoring of the numbers involved.

  63 = 3²·7

  140 = 2²·5·7

The "change of base" formula is also useful.

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Using this, we can write ...

  a = log(3)/log(2)   ⇒   log(3) = a·log(2)

  b = log(5)/log(3)   ⇒   log(5) = b·log(3) = ab·log(2)

  c = log(2)/log(7)   ⇒   log(2) = c·log(7)

This lets us write everything in terms of log(7):

  log(2) = c·log(7)

  log(3) = ac·log(7)

  log(5) = abc·log(7)

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The desired logarithm is ...

_____

The attachment is a numerical check of this result.

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step-by-step explanation:

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step-by-step explanation: