Logarithmic coefficient

Definition

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.

Examples

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 $-0.5$.


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