We can first rewrite the expression as:
#(2^2)^(2x + 1) = 2^14#
Now, we can use this rule of exponents to simplify the term on the left side of the parenthesis:
#(x^color(red)(a))^color(blue)(b) = x^(color(red)(a) xx color(blue)(b))#
#(2^color(red)(2))^color(blue)(2x + 1) = 2^14#
#2^(color(red)(2)color(blue)((2x + 1))) = 2^14#
#2^((color(red)(2)xxcolor(blue)(2x)) + (color(red)(2)xxcolor(blue)(1))) = 2^14#
#2^(4x + 2) = 2^14#
Now, because the bases are the same, for this equation to be equal the exponents must be equal. Therefore, we can write and solve:
#4x + 2 = 14#
#4x + 2 - color(red)(2) = 14 - color(red)(2)#
#4x + 0 = 12#
#4x = 12#
#(4x)/color(red)(4) = 12/color(red)(4)#
#(color(red)(cancel(color(black)(4)))x)/cancel(color(red)(4)) = 3#
#x = 3#
