[tex]25^n-3=5^n+8[/tex]
[tex](5^n)^2-3=5^n+8[/tex], by recognising 25 = 5²
[tex](5^n)^2-5^n-11=0[/tex]
[tex]y^2-y-11=0[/tex], by letting [tex]y=5^n[/tex]
[tex]y= \frac{-1 \pm \sqrt{(-1)^2-(4\times1\times-11)}}{2\times1} = \frac{1 \pm 3\sqrt{5}}{2}[/tex], using the quadratic formula
[tex]y=5^n= \frac{1 + 3\sqrt5}{2} [/tex].
[tex]n = \log_5( \frac{1+3\sqrt{5}}{2} )[/tex], note you could simplify this further using log laws if you are familiar with these.
Notice that [tex] \frac{1-3\sqrt{5}}{2} \ \textless \ 0 [/tex] so this is not a solution as logarithms are only defined for positive values of x.
