The exact solution to the equation is [tex]\rm x=\dfrac{2\ ln\ 2}{3\ ln\ 2-ln\ 5}[/tex].
Given equation,
[tex]2^{(3x-2)} = 5^x[/tex]
Taking log on both sides, we get
[tex]\rm ln\ 2^{(3x-2)}=ln\ 5^x[/tex]
Since [tex]\rm ln\ x^m=m\ ln\ x[/tex], on applying the identity we get,
[tex](3x-2)\rm ln\ 2=x \ ln\ 5[/tex]
using distributive property, we get
[tex]\rm 3x\ ln\ 2-2\ ln\ 2=x\ ln\ 5[/tex]
[tex]\rm 3x\ ln\ 2-x\ ln\ 5=2\ ln\ 2[/tex]
Now, taking x common from L.H.S we get,
[tex]\rm x \ [3\ ln\ 2-ln\ 5]=2\ ln\ 2[/tex]
[tex]\rm x=\dfrac{2\ ln\ 2}{3\ ln\ 2-ln\ 5}[/tex]
Hence the correct option is option A.
For more details on identity of logarithm follow the link:
https://brainly.com/question/7302008
