Question

Solve the following exponential equations.
(10^x)^2 − 2(10^x) − 1 = 0

Answer 1

x = log(1+√2) ≈ 0.38277569

Explanation:

Let z = 10^x. Then the equation can be rewritten ...

z^2 -2z -1 = 0

z^2 -2z +1 -2 = 0 . . . . rewrite to a form with a square

(z -1)^2 = 2 . . . . . . . . . add 2

z -1 = ±√2 . . . . . . . . . . square root

z = 1 ± √2 . . . . . . . . . . add 1

From the definition of z, we know it must be positive, so the only viable solution here is ...

z = 10^x = 1+√2

x = log(1+√2)

x ≈ 0.38277569