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: $$\log_a(n) = \frac{\log_b(n)}{\log_b(a)}$$

Parents: