What is the inverse of the logarithmic function f(x) = log_4(x - 2)?

To determine the inverse of the logarithmic function ( f(x) = \log_4(x - 2) ), we start by rewriting the function in exponential form. The general form of a logarithm tells us that if ( f(x) = \log_b(a) ), then ( a = b^{f(x)} ).

In this case, we can set ( y = f(x) ), leading to the equation:

[ y = \log_4(x - 2) ]

To find the inverse, we switch ( x ) and ( y ):

[ x = \log_4(y - 2) ]

Now we convert this logarithmic equation into its exponential form:

[ y - 2 = 4^x ]

Next, to isolate ( y ), we add 2 to both sides:

[ y = 4^x + 2 ]

This tells us that the inverse function is:

[ f^{-1}(x) = 4^x + 2 ]

This matches with one of the choices, confirming it as the correct answer. The rationale behind this is that the inverse operation effectively “undoes” what the logarithmic function was doing, allowing us