What is the solution for x in the logarithmic equation 2log_2(x - 4) = log_3(4)?

To solve the equation (2\log_2(x - 4) = \log_3(4)), the first step involves rewriting the left-hand side using the power rule of logarithms. This rule states that (a\log_b(M) = \log_b(M^a)), so applying it here transforms the equation to:

[

\log_2((x - 4)^2) = \log_3(4)

]

In order to solve for (x), we need to express both sides of the equation with the same logarithm base or find a way to isolate (x). Since the bases of the logarithms are different, it’s more straightforward to remove the logarithms by exponentiating each side.

To make further progress, observe that to isolate (x), we can relate the two logarithmic expressions. From the right-hand side, we know:

[

4 = 3^{\log_3(4)},

]

which yields:

[

\log_3(4) = \log_3(3^{\log_3(4)}).

]

Now equate the expressions and simplify:

Since we have:

[

(x - 4)^