How do I solve for $t$ in this question? $5^{2t+2}-{100}^{2t}=625$

Question

How do I solve for $t$ in this question?

$5^{2t+2}-{100}^{2t}=625$

I have tried to express the LHS in terms of 2 and 5 but I don't seem to find any useful result. Any help will be appreciated. Thank you.

Answer 1

For $t<0$ we obtain: $$625+100^{2t}>625>25>5^{2+2t}.$$ For $t\geq0$ by AM-GM we obtain: $$625+100^{2t}\geq2\cdot25\cdot10^{2t}>5^{2+2t},$$ which says that our equation has no real roots.

Answer 2

Use $5^{2t+2}=25\cdot 5^{2t}$, then conclude that for $t \gt 0$ the left side is negative and there is no solution. For $t \le 0$ the left side is less than $1$, so there is no solution. Do you have the problem right? Should the minus sign be plus? There is still not a nice solution, but there is one.