Recall that 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: ${log}_{b} (x \cdot y) = l o g_{b} (x) + {log}_{b} (y) .$

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: ${log}_{a} (n) = \frac{{log}_{b} (n)}{{log}_{b} (a)}$