TITLE: What happen to units when we take Ln?
QUESTION [5 upvotes]: What happens to units when we take the Ln of the both side of the equation?
For example for an equation $PV^k = C$ (which becomes $\ln P = -k \ln V + \ln C$), how can we find the units of $k$ and $C$?
-->(the unit of $P$ is mmHg and the unit of $V$ is $cm^3$)
Thank you so much..

REPLY [3 votes]: It should be obvious that in the $PV^k=C$ equation the value of $C$ depends on the units of $P$ and $V$. For example, one can measure volume $V$ is in $m^3$, or liters, or $ft^3$. Then I would write the above equation as $$\frac P{Nm^{-2}}\frac{V^k}{m^{3k}}=\frac{C}{Nm^{3k-2}}$$
Then one can use dimensionless quantities $p=\frac P{Nm^{-2}}$, $v=\frac v{m^3}$, $c=\frac{C}{Nm^{3k-2}}$.
With this, you can take the logarithm
$$\ln p=-k\ln v+\ln c$$