What is the solution of log₃(x²) = 4096?

To solve the equation log₃(x²) = 4096, we start by using the properties of logarithms.

Firstly, we can rewrite this equation in its exponential form. The equation states that the base (3) raised to the power of 4096 is equal to x²:

3⁴⁰⁹⁶ = x²

Next, to isolate x, we take the square root of both sides of the equation:

x = ±√(3⁴⁰⁹⁶)

We can simplify the right side further. Since 3⁴⁰⁹⁶ can be expressed as (3²⁰⁴⁸)², we find that:

x = ±3²⁰⁴⁸

Therefore, the solutions to the original equation are:

x = 3²⁰⁴⁸ and x = -3²⁰⁴⁸