SOLUTION: Solve for x if log x base5+6 log5 base x=5

The starting equation is

     +  = 5.


Change the base   = .


Then the original equation takes the form

     +  = 5.


Introduce new variable y = .
Then the equation takes the form

    y + = 5.


Multiply all the term by y and reduce to quadratic equation

    y^2 - 5y + 6 = 0.


Factor left side

    (y-2)*(y-3) = 0.


The roots are y= 2 and y= 3.


From the root y= 2 we get   = 2,  x =  = 25.

From the root y= 3 we get   = 3,  x =  = 125.


At this point, the problem is just solved in full.


ANSWER.  The given equation has two solutions, x= 25  and  x= 125.