Your m = 5ln(5) is correct, and your equation of the tangent line is correct also.

I found m in another way:

d/dx a^u = ln(a) *a^u *du/dx

So, y = 5^(-1/x)

y' = ln(5) *5^(-1/x) * (1/(x^2))

At point (-1,5),

y' = ln(5) *5^(-1 / -1) *(1 / (-1)^2)

y' = ln(5) *5^1 *1/1

y' = 5ln(5) --------------which is m also.