Logarithm of 1

Theorem

The following formula holds: $$\log(1)=0,$$ where $\log$ denotes the logarithm.

Proof

By the definition, $$\log(z) = \displaystyle\int_1^z \dfrac{1}{\tau} \mathrm{d}\tau.$$ Plugging in $z=1$ and using the integral from a to a, $$\begin{array}{ll} \log(1) &= \displaystyle\int_1^1 \dfrac{1}{\tau} \mathrm{d}\tau \\ &= 0, \end{array}$$ as was to be shown.

References

• 1964: Milton Abramowitz and Irene A. Stegun: Handbook of mathematical functions ... (previous) ... (next): $4.1.12$