With a TI-84 and other technology it is easy to get an approximate answer.

An approximate answer I can read right off my trusty TI-84 calculator as

x = 15.545914

However, the teacher who assigned this problem is probably not teaching a course
on how to use technology to solve math problems.  So I think students who are
assigned problems like this are supposed to get EXACT answers, in terms of
standard functions, such as logarithms and exponential functions. So I'll strive
for an EXACT answer. It's not going to be easy.   

(7^log₈x)(x^log₉x) = 3969, 



The left side is a strictly increasing function.

When x=9 the left side is





Thus x must be greater than 9, since the left side must increase all the way to
3969.





Divide both sides by 72.

 

Take natural logs of both sides:

 





Change to natural logs:



Since ,



let ln(x) = y,  ln(3) = a, and ln(7) = b























Since x > 9 then ln(x) > 2, so we must use the + sign.





Since 



That's the EXACT answer.

When you evaluate that with a calculator, you'll get x = 15.54591432.

Edwin

Thanks, Ikleyn.  You're right, now that I look more closely.  The 8 was so small,
I thought it was a 3.  I may redo it with 8.  But it will probably be harder to
get an exact answer since 8 and 9 are relatively prime. Using 8, the approximate
answer is x = 28.81741

Edwin