# Logarithmic identities

Recall that \(\log_b(n)\) is defined to be the (possibly fractional) number of times that you have to multiply 1 by \(b\) to get \(n.\) Logarithm functions satisfy the following properties, for any base \(b\):

Inversion of exponentials: \(b^{\log_b(n)} = \log_b(b^n) = n.\)

Log of 1 is 0: \(\log_b(1) = 0\)

Log of the base is 1: \(\log_b(b) = 1\)

Multiplication is addition in logspace: \(\log_b(x\cdot y) = log_b(x) + \log_b(y).\)

Exponentiation is multiplication in logspace: \(\log_b(x^n) = n\log_b(x).\)

Symmetry across log exponents: \(x^{\log_b(y)} = y^{\log_b(x)}.\)

Change of base: \(\log_a(n) = \frac{\log_b(n)}{\log_b(a)}\)

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