the math is horrendous, but i used an online quadratic equation calculator to solve to the degree of accuracy required.
the concept is as follows:
start with 3log5(y) - logy(5) = 2
conver to base of 10 log as follows:
log5(y) = log(6)/log(5)
logy(5) = log(5)/log(y)
equation becomes 3log(y)/log(5) - log(5)/log(y) = 2
log(anything) is assumeed to be log10(anything).
your calculator can handle log10(anything) by use of the calculator LOG function.
so your equation has become:
3log(y)/log(5) - log(5)/log(y) = 2
multiply both sides of this equation by log(5)log(y) to get:
3log(y)^2 - log(5)^2 = 2log(5)log(y)
subtract 2log(5)log(y) from both sides of the equation to get:
3log(y)^2 - 2log(5)log(y) - log(5)^2 = 0
let x = log(y).
equation becomes 3x^2 - 2log(5)x - log(5)^2 = 0
this is a quadratic equation that can be solved by using the quadratic formula.
i spared myself the drudgery by using the following online quadratic equation solver:
https://www.mathsisfun.com/quadratic-equation-solver.html
since this solver couldn't handle -2log(5) or -log(5)^2, i had to convert them to fractions.
-2log(5) became -1.397940009
-log(5)^2 became -.48859067
these were input into the quadratic solve and the result was:
x = 0.69897000443178 or x = -0.23299000143178
since x is equal to log(y), then the solution became:
log(y) = 0.69897000443178 or log(y) = -0.23299000143178
log(y) = 0.69897000443178 if and only if 10^0.69897000443178 = y
this led to y = 5.000000001 or y = .5848035477
it remained to place these values into the original equation to see if that equation holds true.
it holds true for y = 5.000000001
it also holds true for y = .5848035477
the solution from the quadratic solver is shown below:
i also solved the equation graphically as shown below:
the graphical solutions are rounded to 3 decimal places but they confirm the alebraic solutuion is correct.
in the graph, i used x instead of y.
same equation with a change in variable name.
the intersection of the 2 graphs is the solution, with the correct value of the original y being the x-coordinate of the coordinate pair.
for example, the coordinate pair of (.585,2) says that the value of x is .585.
you would simply read that as y = .585.
similarly, the coordinate pair of (5,2) says that the value of x is 5.
you would simply read that as y = 5.