The fundamental product rule of logarithms represents a mathematical relation between the logarithm of the product of two or more numbers and the sum of their logarithms.

The property of product law of logarithms is mainly used in mathematics in two cases.

- To shorten sum of logarithms of two or more quantities as logarithm of product of the quantities.
- To expand logarithm of a quantity as sum of logarithms of two or more quantities whose product is equal to the quantity.

Observe each case of using product rule as formula from the following examples.

If two or more sum of logarithmic terms whose bases are same and connected by a plus sign, then sum of logarithms of quantities can be simplify written as logarithm of product of quantities by the product rule of logarithms.

$(1) \,\,\,\,\,\,$ $\log{3} + \log{4}$

$\implies \log{3}+\log{4} \,=\, \log{(3 \times 4)}$

$\implies \log{3}+\log{4} \,=\, \log{12}$

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

$\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, =\,\,$ $\log_{2}{(5 \times 6 \times 7)}$ $\,=\,$ $\log_{2}{210}$

$(3) \,\,\,\,\,\,$ $\log_{e}{8} + \log_{e}{9} + \log_{e}{10} + \log_{e}{11}$

$\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, =\,\,$ $\log_{e}{(8 \times 9 \times 10 \times 11)}$ $\,=\,$ $\log_{e}{7920}$

$(4) \,\,\,\,\,\,$ $\log_{27}{\Bigg(\dfrac{2}{3}\Bigg)}$ $+$ $\log_{27}{\Bigg(\dfrac{4}{5}\Bigg)}$ $+$ $\log_{27}{\Bigg(\dfrac{6}{7}\Bigg)}$

$\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, =\,\,$ $\log_{27}{\Bigg(\dfrac{2}{3} \times \dfrac{4}{5} \times \dfrac{6}{7}\Bigg)}$

$\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, =\,\,$ $\log_{27}{\Bigg(\dfrac{2 \times 4 \times 6}{3 \times 5 \times 7}\Bigg)}$

$\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, =\,\,$ $\require{cancel} \log_{27}{\Bigg(\dfrac{2 \times 4 \times \cancel{6}}{\cancel{3} \times 5 \times 7}\Bigg)}$

$\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, =\,\,$ $\log_{27}{\Bigg(\dfrac{2 \times 4 \times 3}{1 \times 5 \times 7}\Bigg)}$

$\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, =\,\,$ $\log_{27}{\Bigg(\dfrac{24}{35}\Bigg)}$

$(5) \,\,\,\,\,\,$ $\log_{x}{a} + \log_{x}{b} + \log_{x}{c}$

$\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, =\,\,$ $\log_{x}{(a \times b \times c)}$

$\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, =\,\,$ $\log_{x}{abc}$

The product rule of logarithms is also used to expand log of a quantity as sum of the logs of the quantities by converting the quantity as product of two or more quantities.

$(1) \,\,\,\,\,\,$ $\log_{3}{10}$

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

$\implies \log_{3}{10}$ $\,=\,$ $\log_{3}{2} + \log_{3}{5}$

$(2) \,\,\,\,\,\,$ $\log{105}$

$\implies \log{105} \,=\, \log{(3 \times 5 \times 7)}$

$\implies \log{105} \,=\, \log{3} + \log{5} + \log{7}$

$(3) \,\,\,\,\,\,$ $\log_{17}{1430}$

$\implies \log_{17}{1430} \,=\, \log_{17}{(2 \times 5 \times 11 \times 13)}$

$\implies \log_{17}{1430}$ $\,=\,$ $\log_{17}{2}$ $+$ $\log_{17}{5}$ $+$ $\log_{17}{11}$ $+$ $\log_{17}{13}$

$(4) \,\,\,\,\,\,$ $\log_{a}{xy}$

$\implies \log_{a}{xy} = \log_{a}{(x \times y)}$

$\implies \log_{a}{xy} = \log_{a}{x} + \log_{a}{y}$

$(5) \,\,\,\,\,\,$ $\log_{e}{\Bigg(\dfrac{15}{7}\Bigg)}$

$\implies \log_{e}{\Bigg(\dfrac{15}{7}\Bigg)} \,=\, \log_{e}{\Bigg(\dfrac{3 \times 5}{7}\Bigg)}$

$\implies \log_{e}{\Bigg(\dfrac{15}{7}\Bigg)} \,=\, \log_{e} 3 + \log_{e}{\Bigg(\dfrac {5}{7}\Bigg)}$

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