x = Log[150]/Log[216] which is approx. 0.9321629344692847820488219720576457815003942358830985

I hope you set the brackets right in your question.

Well, this would be the answer to your problem:

6^(3x)/15 = 1/15 * 6^(3 x) = 10

Rewrite this equation as by multiplying both sides of the equation with 15:

6^(3 x) = 150
6^(3 x) = 150
(6^3)^x = 150
216^x = 150

=> x = Log[150]/Log[216]

Log of any base would work here, as long as both
Log-Functions have the same base.

=> x = Log[150]/Log[216] which is approx. 0.9321629344692847820488219720576457815003942358830985

I added a small plot for you.

Attachments: 

sol.jpg