Logarithmic coefficient

A factor that appears in logarithmic form in a term is called a logarithmic coefficient.

Introduction

Logarithmic terms are formed generally by the involvement of at least one quantity in logarithmic form. If the quantity in logarithmic form multiplies another quantity to form a logarithmic term, then the quantity in logarithmic form is known as a logarithmic coefficient for the remaining quantity.

Examples

Observe the following examples to understand how to identify logarithmic coefficients in log terms.

$(1) \,\,\,\,\,\,$ $\log_{5}{16}$

The logarithm of $16$ to base $5$ is a log term. It seems there is no other factors involved in forming the term $\log_{5}{16}$ but it is written once. Hence, the meaning of $\log_{5}{16}$ is one time $\log_{5}{16}$.

$\implies \log_{5}{16} \,=\, 1 \times \log_{5}{16}$

$\log_{5}{16}$ is called as logarithm coefficient of $1$.

$(2) \,\,\,\,\,\,$ $-\log_{e}{9}$

Same as the previous case, the negative sign of logarithm of $9$ to base $e$ is written mathematically as follows.

$\implies -\log_{e}{9} \,=\, -1 \times \log_{e}{9}$

$\log_{e}{9}$ is called as logarithmic coefficient of $-1$.

$(3) \,\,\,\,\,\,$ $4{(\log_{2}{619})}^2$

It is a logarithmic term and it can be written as factors in two ways.

$4 \times {(\log_{2}{619})}^2$ and $4\log_{2}{619} \times \log_{2}{619}$

${(\log_{2}{619})}^2$ is logarithmic coefficient of $4$. Similarly, $\log_{2}{619}$ is called logarithmic coefficient of $4\log_{2}{619}$.

$(4) \,\,\,\,\,\,$ $\dfrac{4\log_{7}{11}}{5}$

It is a term in fraction form and it can be written in the following way.

$\implies \dfrac{4\log_{7}{11}}{5} \,=\, \dfrac{4}{5} \times \log_{7}{11}$

Logarithm of $11$ to base $7$ is called logarithmic coefficient of $\dfrac{4}{5}$.

$(5) \,\,\,\,\,\,$ $\dfrac{-0.5}{\log_{a}{b^2}}$

The logarithmic term can also be written in product form by the base swapping log rule.

$\implies \dfrac{-0.5}{\log_{a}{b^2}}$ $\,=\,$ $-0.5 \times \log_{b^2}{a}$

Therefore, $\log_{b^2}{a}$ is called logarithmic coefficient of $-0.5$.