# Logarithmic identities

[summary:

• [ 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]: $\log_b(n) = \frac{\log_a(n)}{\log_a(b)}$]

Recall that [3nd $\log_b(n)$] 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]: $\log_a(n) = \frac{\log_b(n)}{\log_b(a)}$