Proof of the product property

The product property is proved by going back to the definition of logarithm as an exponent.

Proof — Product property

Let $u = \log_a x$ and $v = \log_a y$. Then $a^u = x$ and $a^v = y$. Hence $xy = a^u\cdot a^v = a^{u+v},$ and by the definition of logarithm $\log_a(xy) = u+v = \log_a x + \log_a y$. $\blacksquare$

The quotient and power properties are proved in an entirely analogous way, using $a^{u-v}$ and $(a^u)^k = a^{ku}$ respectively.

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Topics: Logarithmic function
Concepts: Logarithm · Properties of logarithms
Functions: Logarithmic function
Skills: Proving