Like logarithmic terms

The logarithmic terms which contain same logarithmic coefficients are called like logarithmic terms.

Introduction

Logarithm terms are often appeared similar when two or more logarithmic terms are compared. It is possible when the logarithmic terms contain same logarithmic coefficient. Due to similar logarithmic coefficient, the logarithmic terms are called as like logarithmic terms.

Examples

Examine the following examples to identity the like logarithmic terms.

$(1) \,\,\,$ $6\log_{3}{7}$ and $-8\log_{3}{7}$

Express both terms as factors by factorization (or) factorisation method.

$6 \times \log_{3}{7}$ and $-8 \times \log_{3}{7}$

$6$ and $-8$ are different and numbers. $\log_{3}{7}$ is a logarithmic coefficient of $6$ and $-8$ in the both terms. Therefore, $6\log_{3}{7}$ and $-8\log_{3}{7}$ are similar in appearance and known as like logarithmic terms.

$(2) \,\,\,$ $d\log_{a}{xy}$, $\Big(\dfrac{1}{c}\Big)\log_{a}{xy}$ and $0.6\log_{f}{x}\log_{a}{xy}$

Once again, factorize (or) factorise all three logarithmic terms to identity common logarithmic coefficients.

$d \times \log_{a}{xy}$, $\Big(\dfrac{1}{c}\Big) \times \log_{a}{xy}$ and $0.6 \times \log_{f}{x} \times \log_{a}{xy}$

$\log_{a}{xy}$ is a logarithmic coefficient of $d$ in the first term, a logarithmic coefficient of $\dfrac{1}{c}$ in the second term and also a logarithmic coefficient of $0.6\log_{f}{x}$ in the third term.

In this case, the factor $\log_{f}{x}$ is a logarithmic coefficient of $0.6\log_{a}{xy}$ but it is not appeared in remaining two terms. Due to the common involvement of $\log_{a}{xy}$ in all three terms, the three log terms are appeared similar. Hence, the $d\log_{a}{xy}$, $\Big(\dfrac{1}{c}\Big)\log_{a}{xy}$ and $0.6\log_{f}{x}\log_{a}{xy}$ are called as like logarithmic terms.