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Addition of Logarithmic terms

A mathematical operation of adding a logarithmic term to another logarithmic term is called the addition of logarithmic terms. It is also called as the summation of logarithmic terms.

In logarithmic mathematics, there are two types of logarithmic terms. Hence, it is very important to learn how to add two or more like logarithmic terms and also to learn the procedure of adding the unlike log terms.

Adding Like log terms

In this case, the logarithmic factors in the terms are alike. Hence, it is possible to add them as a term but we have to follow some procedure for calculating the sum of the log terms.

For example, $\log_{3}{11}$, $2\log_{3}{11}$ and $3\log_{3}{11}$.

Observe the three terms, you can understand that there is a logarithmic factor in each term commonly. It allows us to add them, same as the addition of algebraic terms.

Now, write all the terms in a row by placing a plus sign between every two log terms for expressing the addition of like log terms.

$\log_{3}{11}$ $+$ $2\log_{3}{11}$ $+$ $3\log_{3}{11}$

$=\,\,\,$ $1 \times \log_{3}{11}$ $+$ $2 \times \log_{3}{11}$ $+$ $3 \times \log_{3}{11}$

Now, take the common logarithmic factor out from each term.

$=\,\,\,$ $\log_{3}{11} \times (1+2+3)$

$=\,\,\,$ $(1+2+3) \times \log_{3}{11}$

$=\,\,\,$ $6 \times \log_{3}{11}$

$=\,\,\,$ $6\log_{3}{11}$

It can be done in one line. Just add the numerical factors directly and then multiply it with the common logarithmic factor for completing the process of adding like logarithmic terms.

$\,\,\,\therefore\,\,\,\,\,\,$ $\log_{3}{11}$ $+$ $2\log_{3}{11}$ $+$ $3\log_{3}{11}$ $\,=\,$ $6\log_{3}{11}$

Therefore, it clears that the sum of the like log terms is equal to the product of the sum of the numerical factors of the terms and common logarithmic factor.

You can also understand the addition of like log terms from the following examples.

$(1)\,\,\,$ $3\log_{e}{4}+4\log_{e}{4} \,=\, 7\log_{e}{4}$

$(2)\,\,\,$ $\log{17}+2\log{17} \,=\, 3\log{17}$

$(3)\,\,\,$ $10\log_{2}{9}+5\log_{2}{9} \,=\, 15\log_{2}{9}$

$(4)\,\,\,$ $\log_{a}{c}+2\log_{a}{c}+3\log_{a}{c}$ $\,=\,$ $6\log_{a}{c}$

$(5)\,\,\,$ $7\log_{50}{2}+6\log_{50}{2}$ $+$ $\log_{50}{2}+5\log_{50}{2}$ $\,=\,$ $19\log_{50}{2}$

Adding Unlike log terms

In this case, the logarithmic factors in the terms are not alike. Hence, it is not possible to add them as a term but we express the sum of them as a mathematical expression.

For example, $\log_{2}{11}$, $2\log_{10}{11}$ and $3\log_{e}{11}$ are three unlike logarithmic terms.

$\log_{2}{11}$ $+$ $2\log_{10}{11}$ $+$ $3\log_{e}{11}$

Due to lack of the common logarithmic factor in the terms, the sum of unlike logarithmic terms is simply written as an expression. Observe the following examples for understanding it much better.

$(1)\,\,\,$ $3\log_{b}{4}+4\log_{e}{4}$

$(2)\,\,\,$ $\log{5}+7\log_{5}{10}$

$(3)\,\,\,$ $4\log_{2}{9}+9\log_{3}{9}$

$(4)\,\,\,$ $\log_{a}{c}+2\log_{b}{d}+3\log_{c}{e}$

$(5)\,\,\,$ $20\log_{7}{40}+4\log_{2}{29}$ $+$ $8\log_{9}{10}+2\log_{3}{11}$

Thus, we can add two or more logarithmic terms mathematically in logarithms.

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