Rules of Logarithms - Examples, Exercises and Solutions

$2\log_82=\log_82^2=\log_84$

$2\log_82+\log_83=\log_84+\log_83=$

$\log_84\cdot3=\log_812$

Where:

$3\log_49=\log_49^3=\log_4729$

y

$8\log_4\frac{1}{3}=\log_4\left(\frac{1}{3}\right)^8=$

$\log_4\frac{1}{3^8}=\log_4\frac{1}{6561}$

Therefore

$3\log_49+8\log_4\frac{1}{3}=$

$\log_4729+\log_4\frac{1}{6561}$

$\log_ax+\log_ay=\log_axy$

$\left(729\cdot\frac{1}{6561}\right)=\log_4\frac{1}{9}$

$\log_49^{-1}=-\log_49$

We break it down into parts

$\log_24=x$

$2^x=4$

$x=2$

$\log_39=x$

$3^x=9$

$x=2$

We substitute into the equation

$\frac{1}{2}\cdot2\log_38+2\log_37=$

$1\cdot\log_38+2\log_37=$

$\log_38+\log_37^2=$

$\log_38+\log_349=$

$\log_3\left(8\cdot49\right)=\log_3392$$x=2$

We break it down into parts

$\log_61296=x$

$6^x=1296$

$x=4$

$\frac{1}{4}\cdot4\cdot\log_6\frac{1}{2}-\log_63=$

$\log_6\frac{1}{2}-\log_63=$

$\log_6\left(\frac{1}{2}:3\right)=\log_6\frac{1}{6}$

$\log_6\frac{1}{6}=x$

$6^x=\frac{1}{6}$

$x=-1$

$\log_ax-\log_ay=\log_a\frac{x}{y}$

$\log_7x^4-\log_72x^2=$

$\log_7\frac{x^4}{2x^2}=3$

$7^3=\frac{x^2}{2}$

We multiply by: $2$

$2\cdot7^3=x^2$

Extract the root

$x=\sqrt{680}=7\sqrt{14}$

$x=-\sqrt{680}=-7\sqrt{14}$

Defined domain

x>0

x+1>0

x>-1

$\log7x+\log\left(x+1\right)-\log7=\log2x-\log x$

$\log\frac{7x\cdot\left(x+1\right)}{7}=\log\frac{2x}{x}$

We reduce by: $7$ and by $X$

$x\left(x+1\right)=2$

$x^2+x-2=0$

$\left(x+2\right)\left(x-1\right)=0$

$x+2=0$

$x=-2$

Undefined domain x>0

$x-1=0$

$x=1$

Defined domain