Solve for x over the real numbers:
49^x = 7^(x^2-15)
Take the natural logarithm of both sides and use the identity log(a^b) = b log(a):
2 log(7) x = log(7) (x^2-15)
Expand out terms of the right hand side:
2 log(7) x = log(7) x^2-15 log(7)
Subtract x^2 log(7)-15 log(7) from both sides:
-(log(7) x^2)+2 log(7) x+15 log(7) = 0
The left hand side factors into a product with four terms:
-(log(7) (x-5) (x+3)) = 0
Divide both sides by -log(7):
(x-5) (x+3) = 0
Split into two equations:
x-5 = 0 or x+3 = 0
Add 5 to both sides:
x = 5 or x+3 = 0
Subtract 3 from both sides:
Answer: | x = 5 or x = -3
