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Product, quotient, power and root

The logarithm of a product is the sum of the logarithms of the numbers being multiplied; the logarithm of the ratio of two numbers is the difference of the logarithms. The logarithm of the p th power of a number is p times the logarithm of the number itself; the logarithm of a p th root is the logarithm of the number divided by p. The following table lists these identities with examples. Each of the identities can be derived after substitution of the logarithm definitions x = b log b x, {\displaystyle x=b^{\log _{b}x},} and/or y = b log b y, {\displaystyle y=b^{\log _{b}y},} in the left hand sides.