Logarithm identities

#[The following identities work for all positive bases $a \neq 1$ and $b \neq 1$, all positive numbers $x$ and $y$, and every real number $r$.][Las siguientes identidades son válidas para todas las bases positivas $a \neq 1$ y $b \neq 1$, todos los números positivos $x$ y $y$, y cada número real $r$.]#

Identity	Examples
$\log_b(xy)$${}= \log_bx + \log_by$ The logarithm of a product equals the sum of the logarithms.	$\log_{5}6$${}= \log_5(3 \times 2)$${}= \log_{5}4 + \log_{5}2$ $\log_b4 + \log_b2$${}= \log(4\times 2)$${}=\log_b8$ $\log_2 12$${}= \log_2(2 \times 2 \times 3)$${}=\log_22 + \log_22 + \log_23$${}= 2\log_22 + \log_23$
$\log_b\left(\dfrac{x}{y}\right) = \log_bx - \log_by$ #[The logarithm of a quotient equals the difference of the logarithms.][El logaritmo de un cociente es la resta de los logaritmos.]#	$\log_5\left(\dfrac{4}{3}\right)$${}=\log_b4 - \log_b3$ $\log_b12 - \log_b3$${}=\log_5\left(\dfrac{12}{3}\right)$${}= \log_b4$ $\log\left(\dfrac{4}{9}\right)$${}=\log4 - \log9$${}=\log4 - (\log3 + \log3)$${}=\log4 - 2\log3$
$\log_b(x^r) = r\log_bx$ The logarithm of an $p$th power is $p$ times the logarithm.	$\log_2(4^3) = 3\log_{2}4$ $4\log_bx = \log_bx^4$ $\log_a(24) = \log_a(2^3 \times 3)$${}=\log_a(2^3) + \log_a 3$${}=3\log_a 2 + \log_a 3$
$\log_bb = 1$ #[and][y]# $\log_b1 = 0$ #[The base $b$ logarithm of $b$ is $1$, and the logarithm of $1$ is $0$.][El logaritmo de $b$ en base $b$ es $1$, y el logaritmo de $1$ es $0$.]#	$\log_{5}5=1$ $\log_61=0$
$\log_b\left(\dfrac{1}{x}\right) = -\log_bx$ The logarithm of a reciprocal is the negative of the logarithm.	$\log_4\left(\dfrac{1}{6}\right) = -\log_{4}5$ $\log\left(\dfrac{5}{3}\right) = \log\left(\dfrac{1}{3/5}\right)$${}=-\log\left(\dfrac{3}{5}\right)$ $-\log 4 = \log 0.25$
$\log_bx = \dfrac{\log_ax}{\log_ab}$ The base $b$ logarithm of $x$ is the logarithm of $x$ over the logarithm of the base.	$\log_47 = \dfrac{\log 7}{\log 4}$ $\log_23 = \dfrac{\log_53}{\log_52}$