Multiplication of Logarithmic terms

A mathematical operation of multiplying a logarithmic term by another logarithmic term is called the multiplication of logarithmic terms.

In logarithmic mathematics, there are two types of logarithmic terms. Hence, it is essential for everyone to learn the process of multiplying the like or unlike logarithmic terms for evaluating their product in mathematics.

Multiplying Like logarithmic terms

In this case, the like logarithmic terms contain the same logarithmic factor. Hence, it is separated from the terms by factoring for expressing it in exponential notation in the product.

For example, $2\log_{7}{13}$, $3\log_{7}{13}$ and $4\log_{7}{13}$.

Look at the three logarithmic terms. It seems they are similar because of their common logarithmic factor. Now, let’s learn how to multiply the like log terms mathematically for getting their product.

Write the like log terms in a row by placing a multiplication sign between every two like logarithmic terms.

$2\log_{7}{13}$ $\times$ $3\log_{7}{13}$ $\times$ $4\log_{7}{13}$

Each logarithmic term can be split into factors and it helps us to separate the common logarithmic factors from the like terms.

$=\,\,\,$ $2$ $\times$ $\log_{7}{13}$ $\times$ $3$ $\times$ $\log_{7}{13}$ $\times$ $4$ $\times$ $\log_{7}{13}$

$=\,\,\,$ $2$ $\times$ $3$ $\times$ $4$ $\times$ $\log_{7}{13}$ $\times$ $\log_{7}{13}$ $\times$ $\log_{7}{13}$

$=\,\,\,$ $(2 \times 3 \times 4)$ $\times$ $(\log_{7}{13}$ $\times$ $\log_{7}{13}$ $\times$ $\log_{7}{13})$

Now, we can express the product of logarithmic factors in exponential notation by exponentiation.

$=\,\,\,$ $(24)$ $\times$ $(\log_{7}{13})^3$

$=\,\,\,$ $24$ $\times$ $(\log_{7}{13})^3$

$=\,\,\,$ $24(\log_{7}{13})^3$

Therefore, it clears that the product of like logarithmic terms is equal to the product of the product of numerical factors and the logarithmic factor raised to the power of the number of logarithmic factors.

The entire multiplication process can be written simply in a single line.

$\therefore\,\,\,$ $2\log_{7}{13}$ $\times$ $3\log_{7}{13}$ $\times$ $4\log_{7}{13}$ $\,=\,$ $24(\log_{7}{13})^3$

The following examples help you to understand the process for the multiplication of like logarithmic terms.

$(1).\,\,\,$ $\log_{2}{9}$ $\times$ $3\log_{2}{9}$ $\,=\,$ $3{(\log_{2}{9})}^2$

$(2).\,\,\,$ $2\log{369}$ $\times$ $(-7)\log{369}$ $\,=\,$ $-14{(\log{369})}^2$

$(3).\,\,\,$ $2\log_{8}{17}$ $\times$ $3\log_{8}{17}$ $\times$ $4\log_{8}{17}$ $\,=\,$ $24{(\log_{8}{17})}^3$

$(4).\,\,\,$ $(-3)\log_{e}{12}$ $\times$ $(-5)\log_{e}{12}$ $\times$ $(-6)\log_{e}{12}$ $\,=\,$ $-90(\log_{e}{12})^3$

$(5).\,\,\,$ $a\log_{a}{b}$ $\times$ $ab\log_{a}{b}$ $\times$ $ac\log_{a}{b}$ $\times$ $bc\log_{a}{b}$ $\,=\,$ $a^3b^2c^2{(\log_{a}{b})}^4$

Multiplying Unlike logarithmic terms

In this case, the unlike logarithmic terms have the different logarithmic factors. Hence, they are separated from the terms by factoring but express them in the product form .

For example, $3\log_{2}{5}$ and $5\log_{10}{9}$

Observe the two logarithmic terms. They are dissimilar due to the different logarithmic factors. Now, let’s learn how to multiply the unlike log terms mathematically for obtaining their product.

Write the unlike log terms in a row by placing a multiplication sign between every two terms.

$3\log_{2}{5}$ $\times$ $5\log_{10}{9}$

Express the terms as multiplying factors to make the multiplication easier.

$=\,\,\,$ $3 \times \log_{2}{5}$ $\times$ $5 \times \log_{10}{9}$

$=\,\,\,$ $3 \times 5$ $\times$ $\log_{2}{5} \times \log_{10}{9}$

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

Multiply the numerical factors and evaluate their product. Similarly, evaluate the product of logarithmic factors but it is not possible. Hence, their multiplication is simply expressed in their product form.

$=\,\,\,$ $15$ $\times$ $\log_{2}{5}\log_{10}{9}$

$=\,\,\,$ $15\log_{2}{5}\log_{10}{9}$

The entire process of multiplication of unlike logarithmic terms can be written directly in a single line.

$3\log_{2}{5}$ $\times$ $5\log_{10}{9}$ $\,=\,$ $15\log_{2}{5}\log_{10}{9}$

The following examples help you to understand the procedure of multiplying two or more unlike logarithmic terms mathematically.

$(1).\,\,\,$ $3\log_{2}{9}$ $\times$ $(-7)\log_{\displaystyle e}{5}$ $\,=\,$ $-21\log_{2}{9}\log_{\displaystyle e}{5}$

$(2).\,\,\,$ $2\log_{5}{11}$ $\times$ $3\log_{7}{20}$ $\,=\,$ $6\log_{5}{11}\log_{7}{20}$

$(3).\,\,\,$ $\log_{3}{17}$ $\times$ $5\log_{4}{7}$ $\times$ $2\log_{5}{11}$ $\,=\,$ $10\log_{3}{17}\log_{4}{7}\log_{5}{11}$

$(4).\,\,\,$ $ab\log_{a}{y}$ $\times$ $bc\log_{b}{x}$ $\,=\,$ $ab^2c\log_{a}{y}\log_{b}{x}$

$(5).\,\,\,$ $(-x)\log_{x}{y^2}$ $\times$ $xy\log_{y}{z^2}$ $\times$ $y\log_{z}{x^2}$ $\,=\,$ $-x^2y^2\log_{x}{y^2}\log_{y}{z^2}\log_{z}{x^2}$

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