mattst88's guide to logarithms

Logarithms

${\log }_2\left(32\right)=x$

This equation simply asks the question 2 raised to what power is 32. It could be rewritten as:

$2^x=\mathrm{32}$

Logarithmic Identities

• ${\log }_b\left(x\right)+{\log }_b\left(y\right)={\log }_b\left(xy\right)$
• ${\log }_b\left(x\right)-{\log }_b\left(y\right)={\log }_b\left(\frac{x}{y}\right)$
• ${\log }_b\left(x^y\right)=y{\log }_b\left(x\right)$
• ${\log }_b\left(\sqrt[y]{x}\right)=\frac{{\log }_b\left(x\right)}{y}$

Change of Base

To find the logarithm in a base not supported on a calculator, b, the following formula can be used.

${\log }_b\left(x\right)=\frac{{\log }_y\left(x\right)}{{\log }_y\left(b\right)}$

Such that

${\log }_3\left(81\right)=\frac{{\log }_{10}\left(81\right)}{{\log }_{10}\left(3\right)}=4$