Properties of Logarithm

Properties of Logarithm (all with $0<a\ne1$, unless stated):

1. $a^{\log_ax}=x$ for $x\in(0,\infty)$, and $\log_a(a^y)=y$ for $y\in R$ (logarithm and exponential undo each other, by definition of inverse function).
2. Product rule. $\log_a(xy)=\log_ax+\log_ay$ for $x,y>0$. Proof: let $\log_ax=u,\log_ay=v,\log_a(xy)=w$; then $a^u=x,a^v=y,a^w=xy$, so $a^w=a^ua^v=a^{u+v}$, giving $w=u+v$.
3. Quotient rule. $\log_a\left(\dfrac xy\right)=\log_ax-\log_ay$ for $x,y>0$. Proof: analogous, using $a^w=\dfrac xy=\dfrac{a^u}{a^v}=a^{u-v}$.
4. Power rule. $\log_ax^r=r\log_ax$ for $x>0,r\in R$. Proof: with $u=\log_ax$, $a^u=x$, so $x^r=(a^u)^r=a^{ru}$, giving $\log_ax^r=ru$.
5. Change of base. $\log_bx=\dfrac{\log_ax}{\log_ab}$, for any valid bases $a,b>0$. Proof: with $v=\log_bx$, $b^v=x$; taking $\log_a$ of both sides, $\log_a(b^v)=\log_ax$, and by the power rule $\log_a(b^v)=v\log_ab$, so $v\log_ab=\log_ax$, i.e. $\log_bx=\dfrac{\log_ax}{\log_ab}$. Named bases. $a=10$ gives the common logarithm, $\log_{10}x$ (written $\log x$ with no base when context is clear in some texts, though this chapter reserves plain $\log$ for base 10 throughout its examples). $a=e$ (irrational, $\approx2.718$, §2.8.3.1) gives the natural logarithm, written $\ln x=\log_ex$; whenever a text writes '$\log x$' with no base at all, in higher mathematics it most often means $\ln x$. $a=2$ gives the binary logarithm $\log_2x$, central to computer science. …