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Division of Logarithmic terms

A mathematical operation of dividing a logarithmic term by another logarithmic term is called the division of logarithmic terms.

In logarithms, there are two types of logarithmic terms. Hence, we see two types of divisions with log terms and it must for everyone to learn the procedure of dividing the like or unlike logarithmic terms in order to find their quotient in mathematics.

Dividing like logarithmic terms

In this case, the like logarithmic terms contain the same logarithmic factor. It causes to eliminate the common logarithmic factor from the quotient in the division of the like log terms. Actually, the common factor from the both terms in the division is eliminated by splitting the like log terms into factors.

For example, $2\log_{3}{7}$ and $4\log_{3}{7}$.

Assume that the log term $4\log_{3}{7}$ is divided by the log term $2\log_{3}{7}$ in this example. The division of them is written in mathematical form as follows.

$\dfrac{4\log_{3}{7}}{2\log_{3}{7}}$

Now, use the factorization (or factorisation) technique to split each term into factors in the division.

$=\,\,\,$ $\dfrac{4 \times \log_{3}{7}}{2 \times \log_{3}{7}}$

$=\,\,\,$ $\dfrac{4}{2}$ $\times$ $\dfrac{\log_{3}{7}}{\log_{3}{7}}$

Now, the logarithmic factors in both terms get cancelled mathematically.

$=\,\,\,\require{cancel}$ $\dfrac{\cancel{4}}{\cancel{2}}$ $\times$ $\dfrac{\cancel{\log_{3}{7}}}{\cancel{\log_{3}{7}}}$

$=\,\,\,$ $2$ $\times$ $1$

$=\,\,\,$ $2$

This mathematical process has proved that the quotient of any two like logarithmic terms is equal to the quotient of their numerical factors. It can be calculated in a single line as follows.

$\therefore \,\,\,$ $\dfrac{4\log_{3}{7}}{2\log_{3}{7}}$ $\,=\,$ $2$

The below listed examples help you to know the procedure for the division of like logarithmic terms.

$(1).\,\,\,$ $\dfrac{-\log_{2}{9}}{3\log_{2}{9}}$ $\,=\,$ $-\dfrac{1}{3}$

$(2).\,\,\,$ $\dfrac{4\log{12}}{\log{12}}$ $\,=\,$ $4$

$(3).\,\,\,$ $\dfrac{2\log_{b}{2}}{8\log_{b}{2}}$ $\,=\,$ $\dfrac{1}{4}$

$(4).\,\,\,$ $\dfrac{-10\log_{e}{7}}{-5\log_{e}{7}}$ $\,=\,$ $2$

$(5).\,\,\,$ $\dfrac{7\log_{4}{5}}{-35\log_{4}{5}}$ $\,=\,$ $-\dfrac{1}{5}$

In this way, we can evaluate the quotient of like logarithmic terms in the mathematics.

Dividing unlike logarithmic terms

Due to the different logarithmic factors in the unlike logarithmic terms, the quotient of the unlike logarithmic terms is also a term in logarithmic form.

For example, $20\log_{2}{9}$ and $5\log_{7}{3}$

Assume that the term $20\log_{2}{9}$ is divided by the term $5\log_{7}{3}$. The division of the unlike logarithmic terms is expressed in the following form.

$\dfrac{20\log_{2}{9}}{5\log_{7}{3}}$

Now, each logarithmic term in both numerator and denominator can be factored for diving one log term by another.

$=\,\,\,$ $\dfrac{20 \times \log_{2}{9}}{5 \times \log_{7}{3}}$

$=\,\,\,$ $\dfrac{\cancel{20} \times \log_{2}{9}}{\cancel{5} \times \log_{7}{3}}$

The logarithmic factors cannot be divided due to the different log factors in the terms but we can divide the numerical factors.

$=\,\,\,$ $\dfrac{4 \times \log_{2}{9}}{1 \times \log_{7}{3}}$

$=\,\,\,$ $\dfrac{4\log_{2}{9}}{\log_{7}{3}}$

This division process of unlike logarithmic terms can be written in one line simply.

$\therefore\,\,\,$ $\dfrac{20\log_{2}{9}}{5\log_{7}{3}}$ $\,=\,$ $\dfrac{4\log_{2}{9}}{\log_{7}{3}}$

The following examples help you to understand the process of dividing the unlike logarithmic terms mathematically.

$(1).\,\,\,$ $\dfrac{2\log_{4}{17}}{5\log_{5}{10}}$ $\,=\,$ $\dfrac{2\log_{4}{17}}{5\log_{5}{10}}$

$(2).\,\,\,$ $\dfrac{14\log_{2}{6}}{2\log_{3}{4}}$ $\,=\,$ $\dfrac{7\log_{2}{6}}{\log_{3}{4}}$

$(3).\,\,\,$ $\dfrac{-9\log{6}}{27\log_{5}{20}}$ $\,=\,$ $-\dfrac{\log{6}}{3\log_{5}{20}}$

$(4).\,\,\,$ $\dfrac{-2\log_{e}{b}}{-20\log_{a}{c}}$ $\,=\,$ $\dfrac{\log_{e}{b}}{10\log_{a}{c}}$

$(5).\,\,\,$ $\dfrac{6\log_{7}{2}}{6\log_{5}{6}}$ $\,=\,$ $\dfrac{\log_{7}{2}}{\log_{5}{6}}$

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