Thank you to everyone who has submitted solutions to this problem.
Here are Sergio from Kings College of Alicante's solutions to these equations.
Kathryn from Sandbach High School has also solved these equations, and has reflected on what methods she used:
I used a variety of methods to solve the equations including logs to a base, taking natural logs. Some required rearranging, others substituting to find a quadratic. The graph of $y=(5^{2x})−(5^x)−6$ was rather unusual. The 2 equations similar to $5^x +4^x =8$ were the equations I couldn't solve exactly.
Can anyone think of a nice way of grouping these equations (perhaps in a venn diagram) according to methods used to solve them?