Solve aor $$ x $$ in the follong equations. $$ x^{x}=8 $$

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We wish to solve

xˣ = 8.

A common method for equations where the variable appears both in the base and the exponent is to take logarithms. Here’s one way to proceed:

1. Write the equation as
  xˣ = 8        (1)

2. Taking the natural logarithm of both sides gives
  ln(xˣ) = ln(8).

3. Using the logarithm power rule (ln(aᵇ) = b·ln(a)), we have
  x · ln(x) = ln(8).
Since 8 = 2³, it follows that ln(8) = 3 ln2. Hence, we have
  x · ln(x) = 3 ln2.      (2)

4. At this point it is useful to make the substitution
  u = ln(x) ⟹ x = eᵘ.
Then equation (2) becomes
  eᵘ · u = 3 ln2.
This equation is of the form
  u · eᵘ = constant,
which is exactly the type of equation solved by the Lambert W function. Recall that the Lambert W function, W(z), is defined as the function satisfying
  W(z) e^(W(z)) = z.

5. Compare
  u · eᵘ = 3 ln2    with  W(3 ln2) e^(W(3 ln2)) = 3 ln2.
Thus, we can identify
  u = W(3 ln2).

6. But u was defined as ln(x), so
  ln(x) = W(3 ln2).
Taking exponentials of both sides then gives
  x = exp(W(3 ln2)).

There is also an equivalent way to write the solution. Recall the property
  exp(W(z)) = z/W(z).
Thus, we may also write
  x = (3 ln2) / W(3 ln2).

This is the exact solution in closed form.

To summarize, the solution for x in the equation xˣ = 8 is:
  x = exp(W(3 ln2))
    or equivalently,
  x = (3 ln2) / W(3 ln2).

Because the Lambert W function is not generally an elementary function, this answer is considered fully acceptable. Numerical evaluation will yield approximately x ≈ 2.39, which indeed satisfies xˣ ≈ 8.
The solution to the equation $$ x^{x} = 8 $$ is: $$ x = \exp(W(3 \ln 2)) $$ or equivalently, $$ x = \frac{3 \ln 2}{W(3 \ln 2)} $$ where $$ W $$ is the Lambert W function.

Solve aor \( x \) in the follong equations. \( x^{x}=8 \)

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