Subtraction of Logarithmic terms

A mathematical operation of subtracting a logarithmic term from another logarithmic term is called the subtraction of logarithmic terms.

In logarithms, there are two types of logarithmic terms. Hence, we have to know how to subtraction the like log terms and also to learn the process of subtracting the unlike logarithmic terms.

Subtracting Like logarithmic terms

In some cases, the logarithmic factors in the terms are the same. So, it is possible to subtract them as a term but there is a simple process for evaluating the difference of the like logarithmic terms.

For example, $7\log_{5}{6}$ and $4\log_{5}{6}$ are two like logarithmic terms.

Subtract the $4\log_{5}{6}$ from $7\log_{5}{6}$ and it is expressed in mathematical form by displaying a minus sign between them as follows.

$7\log_{5}{6}-4\log_{5}{6}$

$=\,\,\,$ $7 \times \log_{5}{6}-4 \times \log_{5}{6}$

There is a common logarithmic factor in the both terms of the expression and it is log of $6$ to base $5$. So, we can take out the common factor from them.

$=\,\,\,$ $\log_{5}{6} \times (7-4)$

$=\,\,\,$ $(7-4) \times \log_{5}{6}$

$=\,\,\,$ $3\log_{5}{6}$

The subtraction process can be done in only one simple step as follows.

$\therefore\,\,\,$ $7\log_{5}{6}-4\log_{5}{6}$ $\,=\,$ $3\log_{5}{6}$

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

You can also observe the following examples to understand the subtraction of like logarithmic terms.

$(1). \,\,\,\,\,\,$ $4\log_{5}{6}-7\log_{5}{6}$ $\,=\,$ $-3\log_{5}{6}$

$(2). \,\,\,\,\,\,$ $\log{225}-6\log{225}$ $\,=\,$ $-5\log{225}$

$(3). \,\,\,\,\,\,$ $10\log_{2}{11}-2\log_{2}{11}$ $\,=\,$ $8\log_{2}{11}$

$(4). \,\,\,\,\,\,$ $c\log_{a}{b}-d\log_{a}{b}$ $\,=\,$ $(c-d)\log_{a}{b}$

$(5). \,\,\,\,\,\,$ $9\log_{e}{27}-20\log_{e}{27}$ $\,=\,$ $-11\log_{e}{27}$

Subtracting Unlike logarithmic terms

In this case, the logarithmic factors in the terms are unlike. Hence, it is impossible to subtract them as a term but we write the difference of them as a mathematical expression.

For example, $7\log_{2}{19}$ and $6\log_{5}{22}$ are two unlike logarithmic terms.

Assume that we have to subtract the $6\log_{5}{22}$ from $7\log_{2}{19}$ for obtaining the subtraction of them.

$7\log_{2}{19} \,-\, 6\log_{5}{22}$

The logarithmic factors are different in both terms. Hence, we cannot perform the subtraction. Hence, the subtraction is simplified written as an expression.

You can also understand the subtraction of unlike logarithmic terms from the following examples.

$(1). \,\,\,\,\,\,$ $\log_{2}{5}-4\log_{3}{8}$

$(2). \,\,\,\,\,\,$ $6\log_{7}{25}-8\log{33}$

$(3). \,\,\,\,\,\,$ $f\log_{a}{d}-h\log_{b}{g}$

$(4). \,\,\,\,\,\,$ $8\log_{10}{29}-\log_{20}{15}$

$(5). \,\,\,\,\,\,$ $9\log_{e}{9}-2\log_{\mu}{2}$

Thus, we can subtraction any two logarithmic terms mathematically in logarithms.

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