The logarithmic terms which contain different logarithmic coefficients are called unlike logarithmic terms.

Logarithm terms are appeared differently in some cases when two or more logarithmic terms are compared. It is actually possible when the logarithmic terms contain different logarithmic coefficients. Due to dissimilar logarithmic coefficients, the logarithmic terms are called as unlike logarithmic terms.

Study the following examples to understand how to identity the unlike logarithmic terms mathematically.

$(1) \,\,\,$ $6\log_{3}{7}$ and $4\log_{4}{10}$

Write the terms as factors by factorization (or) factorisation method.

$6 \times \log_{3}{7}$ and $4 \times \log_{4}{10}$

$6$ and $4$ are numerals. $\log_{3}{7}$ and $\log_{4}{10}$ are the logarithmic coefficients of $6$ and $4$ respectively. The two factors in logarithmic form $\log_{3}{7}$ and $\log_{4}{10}$ are not like (different). Therefore, the logarithmic terms $6\log_{3}{7}$ and $4\log_{4}{10}$ are called as unlike logarithmic terms.

$(2) \,\,\,$ $-9\log_{2}{41}$, $-9\log_{3}{51}$ and $-9\log_{4}{61}$

Factorise (or) factorize the three terms.

$-9 \times \log_{2}{41}$, $-9 \times \log_{3}{51}$ and $-9 \times \log_{4}{61}$

$-9$ is a common multiplying factor in all three terms but the unlikeness of logarithmic terms are determined not by the multiplying factors in number form but it is determined by the unlikeness of logarithmic factors.

$\log_{2}{41}$, $\log_{3}{51}$ and $\log_{4}{61}$ are logarithmic coefficients of $-9$ in all three terms and they are different in appearance. Therefore, $-9\log_{2}{41}$, $-9\log_{3}{51}$ and $-9\log_{4}{61}$ are called as unlike log terms.

Remember, the numerical coefficient or any other coefficient is not considered to determine the unlikeness of logarithmic terms but logarithmic coefficients are only considerable factors to determine the unlike logarithmic terms.

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