Why does log base b of b raised to the power of n always equal n?

Question

Why does log_bbⁿ always equal n?

Answer 1

Since a log is just an exponent, when you see the expression log_b bⁿ, the entire expression is referring to, or describing, the log ( or exponent) that "b"( i.e. the subscript ) must be raised to in order to produce the final value of bⁿ. So think of it this way: b^? = bⁿ. Since the bases are both "b", the unknown log ( or exponent) must be "n".

Answer 2

Recall that

log_bb = 1

Therefore,

log_b(bⁿ) = n log_bb = n(1) = n