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.

It is mainly used possibly in two cases in mathematics.

- To express sum of log of two or more numbers as the logarithm of product of them.
- To expand log of product of two or more numbers as sum of their logs.

Observe the following basic examples to understand how to use product law in logarithmic mathematics.

If base of two or more logarithmic terms is same and connected by a plus sign, then just write logarithm of product of them to same base.

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

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

$\implies \log 3 + \log 4 = \log 12$

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

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

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

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

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

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

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

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

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

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

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

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

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

The logarithm of a number can be expanded as the sum of the logarithm of each factor of the number by writing the number as two or more multiplying factors.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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