PLEASE HELP A-level C2 maths logs homework

Can somebody please help me with this maths question, it is asking me to find the value of p but I'm unsure how to do that. any help would be welcome, with an explanation too if that is okay. Thank you

Can somebody please help me with this maths question, it is asking me to find the value of p but I'm unsure how to do that. any help would be welcome, with an explanation too if that is okay. Thank you

If you have $\displaystyle 2^{f(x)} = 2^{f(x)}$ then you can ascertain that $g(x) = f(x)$, but in this case, obviously both sides are not powers of 2. So you need to make them into that. Notice that $4 = 2^2$, so you can write:

$\displaystyle 2^{p+1} = 4^{p-2} \iff 2^{p+1} = \left(2^2\right)^{p-2} \iff 2^{p+1} = 2^{2(p-2)}$ and hence, we know that: $p+1 = 2(p-2)$ from which it should be easy to solve for $p$.

For the second one, try and do it yourself, it's the same method, except we write $125 = 5^3$.

For part c) 4 is 2 squared. Is that a clue? Can you rearrange the rhs now?

Look for something similar in d)

thank you so much for your help, I can do it now