We have: #2 log_(6)(4) - frac(1)(4) log_(6)(16) = log_(6)(x)#
Let's begin this by expressing #- frac(1)(4) log_(6)(16)# in terms of an argument of #4#:
#Rightarrow 2 log_(6)(4) - frac(1)(4) log_(6)(4^(2)) = log_(6)(x)#
#Rightarrow 2 log_(6)(4) - frac(2)(4) log_(6)(4) = log_(6)(x)#
#Rightarrow 2 log_(6)(4) - frac(1)(2) log_(6)(4) = log_(6)(x)#
Then, we can subtract the like terms on the left-hand side of the equation:
#Rightarrow (2 - frac(1)(2))log_(6)(4) = log_(6)(x)#
#Rightarrow frac(3)(2) log_(6)(4) = log_(6)(x)#
#Rightarrow log_(6)(4^(frac(3)(2))) = log_(6)(x)#
Now, both sides of the equation are in terms of the logarithm of base #6#.
We can cancel these logarithms out by exponentiating both sides by #6#:
#Rightarrow 6^(log_(6)(4^(frac(3)(2)))) = 6^(log_(6)(x))#
#Rightarrow 4^(frac(3)(2)) = x#
#therefore x = 8#
