Question 278787
your equations reads like this:


{{{log(3,(2x-1)) - log(3,(x-4)) = 2}}}


since {{{log(x/y) = log(x) - log(y)}}}, your equation becomes:


{{{log(3,((2x-1)/(x-4))) = 2}}}


The basic definition of logarithms states that {{{y = log(b,x)}}} if and only if {{{b^y = x}}}.


by the basic definition of logarithms, {{{log(3,((2x-1)/(x-4))) = 2}}} if and only if {{{3^2 = (2x-1)/(x-4)}}}


simplify to get:


9 = (2x-1)/(x-4)


multiply both sides of this equation by (x-4) to get:


9*(x-4) = (2x-1)


simplify to get:


9x - 36 = 2x - 1


subtract 2x from both sides of this equation and add 36 to both sides of this equation to get:


9x - 2x = -1 + 36


simplify to get:


7x = 35


divide both sides of this equation by 7 to get:


x = 5


your answer should be x = 5.


substitute in your original equation to see if this is true.


your original equation is:


{{{log(3,(2x-1)) - log(3,(x-4)) = 2}}}


substitute 5 for x to get:


{{{log(3,(2*5-1)) - log(3,(5-4)) = 2}}} which becomes:


{{{log(3,9) - log(3,1) = 2}}} which becomes:


{{{log(3,(9/1)) = 2}}} which becomes:


{{{log(3,9)}}} = 2 which is true if and only if {{{3^2 = 9}}} which it is.


your answer is:


x = 5