Difference between revisions of "Logarithm of 1"
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where $\log$ denotes the [[logarithm]]. | where $\log$ denotes the [[logarithm]]. | ||
==Proof== | ==Proof== | ||
| + | By the definition, | ||
| + | $$\log(z) = \displaystyle\int_1^z \dfrac{1}{\tau} \mathrm{d}\tau.$$ | ||
| + | Plugging in $z=1$ and 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== | ==References== | ||
Revision as of 12:24, 17 September 2016
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 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): $4.1.12$
