Wiskunde Multiple Choice Vraag uit Gricha's Wiskundige Vragenbank

$https://latex.codecogs.com/png.image?\inline \large \dpi{120}\log \frac 21 +\log \frac 32 +\log \frac 43 +\cdots +\log \frac {99}{98} +\log \frac {100}{99} =$	A. 1
	B. 2
	C. 10
	D. 100
	E. een niet geheel getal

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Translation in E N G L I S H

$https://latex.codecogs.com/png.image?\inline \large \dpi{120}\log \frac 21 +\log \frac 32 +\log \frac 43 +\cdots +\log \frac {99}{98} +\log \frac {100}{99} =$	A. 1
	B. 2
	C. 10
	D. 100
	E. none of the above

Oplossing - Solution

1^ste manier :
$log{\frac{2}{1}}+log{\frac{3}{2}}+log{\frac{4}{3}}+\ldots+log{\frac{99}{98}}+log{\frac{100}{99}}=log{\frac{2.3.4....99.100}{1.2.3...98.99}}=log{\;}100=2$
2^de manier :
$\log{\frac21}+\log{\frac32}+\log{\frac43}+\ldots+\log{\frac{99}{98}}+\log{\frac{100}{99}}\\=(\log{2}-\log{1})+(\log3-\log2)+(\log4-\log3)+\!\ldots\!+(\log99-\log98)+\log(100-\log99)\\=\log2+\log3-\log2+\log4-\log3+\ldots+\log99-\log98+\log100-\log99\\=\log100=2$
(alle termen vallen weg, behalve log 100)

GWB