Answer:
x ≈ 2.467
Explanation:
You want the solution to 5^(3x -2) = 7^(x +2).
Logs
Logarithms turn an exponential problem into a linear problem. Taking logs, we have ...
(3x -2)·log(5) = (x +2)·log(7)
x(3·log(5) -log(7)) = 2(log(7) +log(5)) . . . . . separate variables and constants
x = log(35²)/log(5³/7) = log(1225)/log(125/7) . . . . divide by x-coefficient
x ≈ 2.46693
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Additional comment
A graphing calculator can solve this nicely as the x-intercept of the function f(x) = 5^(3x-2) -7^(x+2). Newton's method iteration is easily performed to refine the solution to calculator precision.