[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)}$~$