# Logarithmic Term

## Definition

A term that contains a logarithmic expression is called a logarithmic term.

Any quantity can be expressed in logarithmic form. If a quantity is expression in the form a term and it contains a logarithmic form expression, then the term is called a logarithmic term.

For example, $3$ is a number. It can be expression in logarithmic form.

$3 \,=\, \log_{2}{8}$

Mathematically, the term $\log_{2}{8}$ can be written instead of $3$.

Basically, logarithm of $8$ to the base $2$ is a term and it represents a number. Moreover, the term is in the logarithmic form. Hence, the term $\log_{2}{8}$ is called a logarithmic term.

### Examples

Logarithmic terms are formed in three different types possibly.

01

#### Direct form

Only a single logarithmic term represents the quantity completely.

$(1) \,\,\,\,\,\,$ $\log_{3}{10}$

$(2) \,\,\,\,\,\,$ ${(\log_{6}{1898})}^4$

$(3) \,\,\,\,\,\,$ $\log_{e}{91}$

$(4) \,\,\,\,\,\,$ $\log_{a}{b^2}$

$(5) \,\,\,\,\,\,$ $\log_{xy}{(1+xyz)}$

02

#### Product form

The terms are also formed by the product of numbers and logarithmic form expressions. Due to the involvement of the logarithmic form expressions, the terms are called logarithmic terms.

$(1) \,\,\,\,\,\,$ $5\log_{2}{7}$

$(2) \,\,\,\,\,\,$ $-8{(\log_{4}{190})}^2$

$(3) \,\,\,\,\,\,$ $0.78\log_{e}{11211}$

$(4) \,\,\,\,\,\,$ $b\log_{c}{ac^3}$

$(5) \,\,\,\,\,\,$ $(2+x^2)\log_{z}{(1-x^2)}$

03

#### Division form

The terms are also formed in division form to represent quantities by their quotients. If a term contains, a logarithmic form expression, then it is known as a logarithmic term.

$(1) \,\,\,\,\,\,$ $\dfrac{-7}{\log_{5}{3}}$

$(2) \,\,\,\,\,\,$ $\dfrac{{(\log_{12}{50})}^7}{10}$

$(3) \,\,\,\,\,\,$ $\dfrac{5}{0.9\log_{e}{(7g)}}$

$(4) \,\,\,\,\,\,$ $\dfrac{\log_{10}{(xyz)}}{z^2}$

$(5) \,\,\,\,\,\,$ $\dfrac{1-b}{\log_{b}{(1-ab^8)}}$