Log­a­r­ith­mic identities

Recall that ${\log }_b\left(n\right)$ 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: $b^{{\log }_b\left(n\right)}={\log }_b\left(b^n\right)=n.$

• Log of 1 is 0: ${\log }_b\left(1\right)=0$

• Log of the base is 1: ${\log }_b\left(b\right)=1$

• Multiplication is addition in logspace: ${\log }_b(x\cdot y)=lo{g}_b\left(x\right)+{\log }_b\left(y\right).$

• Exponentiation is multiplication in logspace: ${\log }_b\left(x^n\right)=n{\log }_b\left(x\right).$

• Symmetry across log exponents: $x^{{\log }_b\left(y\right)}=y^{{\log }_b\left(x\right)}.$

• Change of base: ${\log }_a\left(n\right)=\frac{{\log }_b\left(n\right)}{{\log }_b\left(a\right)}$