Why are exponents dimensionless?

Question

Like: P(h)=P₀e^−gh/A. The −gh/A part of the equation is said to be dimensionless, why is that?

Answer 1

Exponents are always dimensions, because the exponential of a unitful quantity is not defined. For example, if you have the quantity g, which we will call the acceleration due to gravity which is equal to 9.8 m/s². Then say you have some distance h. We will let h = 5 m. The product of these quantities is then gh = 9.8 m/s² x 5 m = 49 m²/s². So this quantity gh has units of m2/s2. Now what is (gh)²? This is simply 2,401 m⁴/s⁴. Notice, that the units have changed. Now what is exp(gh)? We now have to compute exp(49) (this is fine), but what is the exp(m²/s²)? This is undefined. This is why quantities in the exponential or logarithmic function are always dimensionless or unitless.

Hope this helps!