Three numbers, one relationship: base^exponent = result
2^5 = 32 connects three numbers: the base (2), the exponent (5), and the result (32). Each of exponentiation, roots, and logarithms starts from two of these three numbers and solves for the missing one.
Exponentiation: given base and exponent, find the result
2^5 = 32 — base and exponent known, solving for the result.
Roots: given exponent and result, find the base
The 5th root of 32 asks "what base, raised to the 5th power, gives 32?" The answer is exactly 2 — the nth root calculation undoes exponentiation by solving for the base instead of the result.
Logarithms: given base and result, find the exponent
log₂(32) asks "2 raised to what power gives 32?" The answer is exactly 5 — the logarithm undoes exponentiation a different way, solving for the exponent instead of the base.
A second example, confirming the pattern
3^4 = 81. The 4th root of 81 is exactly 3 (recovering the base). log₃(81) is exactly 4 (recovering the exponent). Every one of these three calculators, given the right two pieces of information, always agrees with what the third one would show — because they're three views of the exact same underlying relationship.