The Product of Logarithm of A and Logarithm of B

Introduction

Recall that we define the logarithm of base 10 of x is the exponent needed to produce x. The equation

\log_{10}x = c means 10^c = x, where x >0.

 

Logarithms to the base 10 are called common logarithms. Most times, \log_{10}x is written as \log x. In this post, we prove that

\log AB = \log A + \log B.

Theorem

\log AB = \log A + \log B.

Proof

Let \log A = x and let \log B = y. Using the definition above, we have AB = (10^x)(10^y).

By the law of exponents, AB = 10^{x + y}.

Getting the logarithm of each side, \log AB = \log 10^{x+y}.

By the definition we have mentioned above, \log AB = x + y.

Substituting \log A to x and substituting \log B to y, we have

\log AB = \log A + \log B. \blacksquare

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