Start with the equation:

\(\ln(x^4 - 4) = 4 \ln x - \ln 4\)

Use the logarithm property \(\ln a - \ln b = \ln \left( \frac{a}{b} \right)\):

\(\ln(x^4 - 4) = \ln \left( \frac{x^4}{4} \right)\)

Since the logarithms are equal, set the arguments equal:

\(x^4 - 4 = \frac{x^4}{4}\)

Multiply through by 4 to clear the fraction:

\(4(x^4 - 4) = x^4\)

Expand and simplify:

\(4x^4 - 16 = x^4\)

Rearrange to form a quadratic equation:

\(3x^4 = 16\)

Divide by 3:

\(x^4 = \frac{16}{3}\)

Take the fourth root of both sides:

\(x = \left( \frac{16}{3} \right)^{1/4}\)

Calculate the value:

\(x \approx 1.52\)