How do you know that this equation is true without just using a calculator for both sides

e^2 = 9^(1/ln(3))

Take the ln of both sides:
ln(e^2) = ln(9^1/ln(3))
Simplify the left hand side:
2 = ln(9^1/ln(3))
Take the power of the right hand side and bring it down in front of the ln
2 = 1/ln(3) * ln(9)
Simplify the fraction:
2 = ln(9) / ln(3)
Turn ln(9) into ln(3^2):
2 = ln(3^2) / ln(3)
Take the power and bring it in front of the ln
2 = 2ln(3) / ln(3)
Cancel the ln(3)
2 = 2
True!

Take the log with base 9 of both sides of the equation.
Assuming all logs written below are of base 9.

You'll get log(e^2) = 1/ln(3)
Using the change of base formula,
log_a (b) = log_c (b) / log_c(a)

log(e^2) = ln(e^2)/ln(9) = 2/2ln(3) = 1/ln(3)