How do I decide the dimensions of a trigonometric quantity and a logarithmic quantity?
For example, what are the dimensions for: $$\frac{C}{B} = \frac{D^2}{A} + \log \left(\frac{AC}{BD}\right)$$
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1$\log , \sin, \cos, e^x ,...$ don't have any dimensions, they are pure numbers, like $1$ – Ofek Gillon Mar 08 '17 at 05:50
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Please use mathjax to format mathematical expressions. To learn more about mathjax, please read MathJax basic tutorial and quick reference. – Yashas Mar 08 '17 at 05:51
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Possible duplicates: http://physics.stackexchange.com/q/13060/2451 and links therein. – Qmechanic Mar 08 '17 at 06:30
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Dimension of any mathematical function is, simply stated, 1 (or more appropriately $M^0L^0T^0$). Also their arguments are supposed to be dimensionless.
So,
$$ [\ln(x)] = M^0\ L^0\ T^0\ K^0\ A^0\ mol^0\ cd^0 = 1 $$ and $$ [x] = M^0\ L^0\ T^0\ K^0\ A^0\ mol^0\ cd^0 = 1 $$
so in your question. $$ \left[\frac{C}{B}\right] = \left[\frac{D^2}{A}\right] = \left[\log(...)\right] = 1 $$ and anything inside must be dimensionless. $$ \left[\frac{AC}{BD}\right] = 1 $$
The Imp
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