Unit Analysis: Exponential & Logarithmic Formulas

In summary, the conversation discusses the application of units in different types of formulas and equations. It is noted that in linear formulas, the resulting unit is proportional to the individual units, but in nonlinear formulas, it may not be the case. It is also mentioned that in formulas involving exponential or logarithmic functions, the resulting quantity must be dimensionless.
  • #1
KFC
488
4
Hi all,
I have a general question about the unit in formula or equation. In some formula like ##F=md^2x/dt^2## or thermal radiation law ##P \propto A\cdot T^4##, if we plug in the unit for each quantity, the resulting unit of the output is the resulting algebra of the units. For example

$$[F] = \text{kg}\cdot\text{m}^2/\text{s}^2$$

In this case, we can say the unit for the force if kg.m^2/s^2, but what happens if the formula is not linear, for example, if there is a formula ##F = \exp(xy)##. I know this formula might not exist in physical world but if it happens to have that and if x and y is not dimensionless, does it mean the unit for F will be exponential? If not, why is that? Why the linear formula will give resulting unit proportional to the individual unit but when the formula becomes nonlinear, they won't give the resulting unit the same way?

Ok, I know that it doesn't have unit like exp(m/t). So does it mean whenever I have formula in exponential or logarithm, the resulting quantity must be dimensionless?
 
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  • #2
KFC said:
So does it mean whenever I have formula in exponential or logarithm, the resulting quantity must be dimensionless?

The quantity inside the exponential or the logarithm must be dimensionless, yes.
 
  • #3
KFC said:
Ok, I know that it doesn't have unit like exp(m/t). So does it mean whenever I have formula in exponential or logarithm, the resulting quantity must be dimensionless?
Yes, the exponent must be dimensionless, but such formulae can be of the form Cexp(something) where C has got the appropriate dimension.
 

1. What is unit analysis?

Unit analysis is a process used in science and mathematics to check the consistency and accuracy of units in a calculation. It involves converting all units to a common unit and checking that they cancel out correctly.

2. How do exponential formulas relate to unit analysis?

Exponential formulas involve using exponents to represent repeated multiplication of a number. In unit analysis, exponents are used to represent repeated units, making it easier to convert between different units.

3. What is the purpose of using logarithms in unit analysis?

Logarithms are used in unit analysis to simplify complex calculations involving large numbers. By taking the logarithm of a number, it can be converted into a more manageable form for calculation.

4. Can unit analysis be used in all areas of science?

Yes, unit analysis can be applied to any scientific calculation where units are involved, such as physics, chemistry, and biology. It is an important tool in ensuring the accuracy and consistency of scientific data.

5. Are there any limitations to unit analysis?

Unit analysis can become more complex when dealing with non-linear equations or when units are raised to non-integer powers. It also does not take into account the significance of quantities, only their magnitude, so it is important to be aware of this when using unit analysis in scientific calculations.

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