Logarithmic Terms

A term that represents a quantity in logarithmic form is called a logarithmic term. It can also be simply called as a log term.

Introduction

Any quantity can be expressed in logarithmic form. If the quantity is written as a term in logarithmic form then the term is known as a logarithmic term.

Example

$3$ is a number. It can be written in logarithmic form as follows.

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

Mathematically, the term $\log_{2}{8}$ represents the quantity $3$ and moreover, it is in logarithmic form. Therefore, the term $\log_{2}{8}$ is called as a logarithmic term, or simply a log term.

Cases

Logarithmic terms are formed in four different ways possibly.

Numbers

Every real number can be expressed in logarithmic form. So, just consider every real number as a logarithmic term.

Direct form

Only a single logarithmic term represents the quantity completely.

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

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

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

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

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

Product form

The product of two or more quantities is also a quantity. So, a term can be a product of two or more quantities in which at least a quantity can be in logarithmic form. The terms are called as log terms in such cases.

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

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

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

$(4) \,\,\,\,\,\,$ $b\sin{(d^2)}\log_{c}{ac^3}$

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

Division form

The quotient of two quantities is also a quantity. So, a term is also quotient of quantities in which at least a quantity can be in log form, then the terms are called as log terms mathematically.

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

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

$(3) \,\,\,\,\,\,$ $\dfrac{5}{0.9\log_{e}{(7g)}\log_{2}{h}}$

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

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