**Introduction**

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

means , where .

Logarithms to the base 10 are called common logarithms. Most times, is written as . In this post, we prove that

.

**Theorem**

.

**Proof**

Let and let . Using the definition above, we have .

By the law of exponents, .

Getting the logarithm of each side, .

By the definition we have mentioned above, .

Substituting to and substituting to , we have

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