Solve log x = (log x^2)^2 base 4

Question

How do you simplify (log x^2)^2 base 4?

Answer 1

log₄x = (log₄(x²))²

 

log₄x = (2log₄x)²

 

log₄x = 4(log₄x)²

 

Let u = log₄x.  Then, in terms of u, the equation is u = 4u²

 

So, u - 4u² = 0

 

     u(1 - 4u) = 0

 

     u = 0 or u = 1/4

 

Since u = log₄x, we have:  log₄x = 0 or log₄x = 1/4

 

If log₄x = 0, then x = 4⁰ = 1

 

If log₄x = 1/4, then x = 4^1/4 = (2²)^1/4 = 2^1/2 = √2

 

The solutions are 1 and √2.