Laws of Logarithms
Laws of Logs
(\(\log_{a}n=x\)\)
(\(a^x=n\)\)
Tip
Addition/Multiplication law: \(\(\log_{a}x+\log_{a}y=\log_{a}xy\)\)
Subtraction/Division Law:\(\(\log_{a}x-\log_{a}y=\log_{a} \frac{x}{y}\)\)
Power Law: \(\(\log_{a}(x^k)=k\log _a(x)\)\)
- \(\log_{a} \frac{1}{x}=\log_{a}x^{-1}=-\log _ax\)
- \(\log_{a}a=1\)
- \(\log_{a}1=0\)
Some log examples
- \(\log_{3}6+\log_{3}7=\log_{3}42\)
- \(\log_{2}15-\log_{2}3=\log_{2}5\)
- \(2\log_{5}3+3\log_{5}2=\log_{5}3^2+\log_{5}2^3=\log_{5}9+\log_{5}8=\log_{5}72\)
- \(\log _{a}(x^2y2^3)=\log_{a}x^2+\log_{a}y+\log_{a}2^3=2\log_{a}x+\log_{a}y+3\log_{a}2\)
- \(\log_{a}\left( \frac{x\sqrt{ y }}{2} \right)=\log_{a}x+\log_{a}\sqrt{ y }-\log_{a}2\)
- \(\log_{10}4+2\log_{10}x=2\)
\(\log_{10}4+\log_{10}x^2=2\)
\(\log_{10}(4x^2)=2\)
\(4x^2=10^2\)
\(x^2=25\)
\(x=5\)
- \(\log_{3}(x+11)-\log_{3}(x-5)=2\)
\(\log_{3} \frac{(x+11)}{(x-5)}=2\)
\(\frac{x+11}{x-5}=3^2\)
\(x+11=9x-45\)
\(56=8x\)
\(x=7\)