Convert bases to be identical

Express both sides with the base 2. Since $16 = 2^{4}$, rewrite the right side as $(2^{4})^{(x-1)}$.

$2^{(x+3)} = (2^{4})^{(x-1)}$

Apply power of a power rule

Using the rule $(a^{m})^{n} = a^{m imes n}$, multiply the exponents on the right side.

$2^{(x+3)} = 2^{4(x-1)}$

Equate exponents

Since the bases are equal, the exponents must be equal: $x + 3 = 4x - 4$.

$x + 3 = 4x - 4$

Solve for x

Rearrange the equation: $3 + 4 = 4x - x$, which gives $7 = 3x$. Dividing by 3 results in $x = \frac{7}{3}$.

$x = \frac{7}{3}$