I'll go as far as to say simplicity should be the driving motivation, not equality.
Do you mean that if someone earns R, they pay $C*log10(R/P), where P is some poverty level for them and C is a constant around 9?
To find C, you have to sum up the log10(R/P) for everyone (including corporations) in the nation and divide that into what the government needs. There's various ways to calculate what the government needs, perhaps you separate FICA and payroll tax income entirely if you keep that, or perhaps you just look at entire Federal spending, I believe around 3.8 trillion.
But to sum up the log10(R/P), you need income distribution data. Someone else did that very roughly with a spreadsheet, I haven't tried it myself. A few years ago when they did that, they figured 3.7 trillion for spending, and C was something like 8.86.
C would also be a bit higher if you were also trying to pay down the debt over a period of time.
If the tax itself is proportional to the utility function, that has the effect of taking much less utility from people with greater R. I'm also unclear on how your C works out. The largest company in the world makes less than $1 trillion per year, so even if P is $1, log10(R/P) is bounded above by 12. Given a US budget of several trillion USD, that seems to make C multiple orders of magnitude bigger.
Sorry if I'm missing something. That was my original reason for asking for a formula above.
Basically, you take revenue, divide it by the poverty level (he's got six kids, so around $45k - $50k), take the log base 10, and multiply by some constant that would be the same for everyone to match the government spending level - that constant these days is probably around 9 or 10. The result is your fair tax rate.