To solve this equation, we can first simplify the logarithmic expression using the property log(a^n) = n*log(a):

Og 9(6√6-15)^2 + log 27(6√6+15)^3
= log(9^1 (6√6-15)^2) + log(27^1 (6√6+15)^3)
= log((9 (6√6-15))^2) + log((27 (6√6+15))^3)
= log((54√6-135)^2) + log((162√6+405)^3)

Now, we can use the log properties log(a) + log(b) = log(ab) and log(a^b) = b*log(a) to simplify further:

= log((54√6-135)^2 (162√6+405)^3)
= log((54√6-135) (162√6+405))^2

Now, we have:
Og 9(6√6-15)^2 + log 27(6√6+15)^3 = log((54√6-135) * (162√6+405))^2

Since the above expression equals 2, we have:
log((54√6-135)*(162√6+405)) = √2

To solve for (54√6-135)*(162√6+405), we can first expand the product using the distributive property:

(54√6-135)(162√6+405)
= 54√6162√6 + 54√6405 - 135162√6 - 135405
= 54 162 6 + 54 405√6 - 135 162√6 - 135 405
= 54 972 - 135 162√6 - 135 * 405
= 52488 - 21870√6 - 54675

Therefore, the simplified expression is:
52488 - 21870√6 - 54675 = √2

This equation can be solved further to find the value of the unknown in the equation.