Math Doubts

Logarithmic coefficient


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

A coefficient is a multiplying factor in a term. If the coefficient is in the logarithmic form in a term, then it is called logarithmic coefficient.


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

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

The logarithmic of $16$ to base $5$ is a term. It seems there is no other factors involve in forming the term with $\log_{5}{16}$ 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 logarithmic coefficient of $1$.

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

Same as the previous case, the 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 logarithmic coefficient of $-1$.

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

It is a multiplying factor and it can be written in two different forms. Hence, it contains two logarithmic coefficients.

$4 \times {(\log_{2}{619})}^2$ 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 change base logarithmic rule in reciprocal form.

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

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

Follow us
Email subscription
Math Doubts
Math Doubts is a best place to learn mathematics and from basics to advanced scientific level for students, teachers and researchers. Know more
Follow us on Social Media
Mobile App for Android users Math Doubts Android App
Math Problems

Learn how to solve easy to difficult mathematics problems of all topics in various methods with step by step process and also maths questions for practising.

Learn more