log_3(3/4+4/5+5/6+6/7+7/8+8/9) divided by log_10(4/5+5/6+6/7+7/8+8/9+9/10)

Note :  3/4+4/5+5/6+6/7+7/8+8/9  = 12611 / 2520

And : 4/5+5/6+6/7+7/8+8/9+9/10 = 12989 / 2520

So we have

log₃  [12611 / 2520 ] / log₁₀ [ 12989/2520]  which we can write as

[log [12611 / 2520] / log [3] ]  /  [ log [ 12989/2520] / log [10] ]       but log10  = 1....so we can ignore this....and we have

[log [12611 / 2520] / log [3] ]  /  [ log [ 12989/2520] ]

Unfortunately, this about all we can do  ......this evaluates to ≈ 2.058