Question

What is a logarithm in simple terms?

Answer 1

Logarithm definition:

• A logarithm is an exponent to which a base must be raised to get a given number.

For example: $$\begin{gathered} 2^5=32 \\ \text{Now for logarithm function} \\ {\log }_2\left(32\right)=5 \end{gathered}$$

• In simple words, the logarithm is the opposite of an exponent.
• The logarithm is of two types: Natural log represented as $\ln$ or ${\log }_e$ and common logarithm represented as ${\log }_{10}$.
• Some properties of logarithm are:

1. ${\log }_b\left(xy\right)={\log }_{b}x+{\log }_{b}y\mathbf{[}\mathbf{x}\mathbf{,}\mathbf{y}\mathbf{>}\mathbf{0}\mathbf{}\mathbf{a}\mathbf{n}\mathbf{d}\mathbf{}\mathbf{b}\mathbf{>}\mathbf{0}\mathbf{,}\mathbf{b}\mathbf{\ne }\mathbf{1}\mathbf{]}$
2. ${\log }_b\left(\frac{x}{y}\right)={\log }_{b}x-{\log }_{b}y\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{[}\mathbf{x}\mathbf{,}\mathbf{y}\mathbf{>}\mathbf{0}\mathbf{}\mathbf{and}\mathbf{}\mathbf{b}\mathbf{>}\mathbf{0}\mathbf{,}\mathbf{b}\mathbf{\ne }\mathbf{1}\mathbf{]}$
3. ${\log }_{b}b=1\mathbf{[}\mathbf{b}\mathbf{>}\mathbf{0}\mathbf{,}\mathbf{b}\mathbf{\ne }\mathbf{1}\mathbf{]}$
4. ${\log }_b{x}^y=y{\log }_{b}x\mathbf{[}\mathbf{x}\mathbf{>}\mathbf{0}\mathbf{}\mathbf{and}\mathbf{}\mathbf{b}\mathbf{>}\mathbf{0}\mathbf{,}\mathbf{b}\mathbf{\ne }\mathbf{1}\mathbf{,}\mathit{y}\mathbf{\in }\mathit{R}\mathbf{]}$
5. ${\log }_{y}x=\frac{{\log }_{b}x}{{\log }_{b}y}\mathbf{[}\mathbf{x}\mathbf{,}\mathbf{y}\mathbf{>}\mathbf{0}\mathbf{,}\mathit{y}\mathbf{\ne }\mathbf{1}\mathbf{}\mathbf{and}\mathbf{}\mathbf{b}\mathbf{>}\mathbf{0}\mathbf{,}\mathbf{b}\mathbf{\ne }\mathbf{1}\mathbf{]}$