Can you take the logarithm of a dimensioned quantity?

No, you can’t. This question has caused some angst in physics forums. Functions such as log , exp , and sin are not defined for dimensioned quantities, and yet you will find expressions such as “log temperature” in physics text books.

Are logarithms dimensionless?

Yes, logarithms always give dimensionless numbers, but no, it’s not physical to take the logarithm of anything with units.

Why can a logarithm only be taken for a Unitless number?

Logarithm of a quantity really only makes sense if the quantity is dimensionless, and then the result is also a dimensionless number. So what you really plot is not log(y) but log(y/y0) where y0 is some reference quantity in the same units as y (in this case y0=1 Volt). Similarly for exp and sin.

What quantities are not dimensionless?

The dimensional formula of moment of inertia is [ML2T0]. This means that angular momentum is not a dimensionless quantity. Therefore, no given quantities are dimensionless.

Does log preserve units?

Any time you’ll have to take a logarithm it would be of a dimensionless quantity; for example the ratio of values of a dimensional quantity. As a result, the logarithm will also be dimensionless; it will have no units. then you simply ignore the units.

What happens to the units when you take the natural log?

I have always casually said, ‘that when one takes the log/ln of a number with units, it becomes unitless’. The real deal is that you cannot take the log (or ln) of a number that actually has units, i.e., before the log (or ln) is applied, the unit must be dimensionless.

Is the dimensionless quantity?

In dimensional analysis, a dimensionless quantity (or more precisely, a quantity with the dimensions of 1) is a quantity without any physical units and thus a pure number….List of dimensionless quantities.

How do logarithms affect units?

The units of a ln(p) would generally be referred to as “log Pa” or “log atm.” Taking the logarithm doesn’t actually change the dimension of the argument at all — the logarithm of pressure is still pressure — but it does change the numerical value, and thus “Pa” and “log Pa” should be considered different units.

What is non-dimensional parameter?

Non-dimensional parameters are routinely used to classify different flow regimes. We propose a non-dimensional parameter, called Aneurysm number (An), which depends on both geometric and flow characteristics, to classify the flow inside aneurysm-like geometries (sidewalls and bifurcations).

Which of the following physical quantities has neither dimensions nor unit?

Relative density is the ratio of two like quantities. Therefore, it has neither unit nor dimensions.

How does natural log effect units?

What happens when you take log of a log?

The laws apply to logarithms of any base but the same base must be used throughout a calculation. This law tells us how to add two logarithms together. Adding log A and log B results in the logarithm of the product of A and B, that is log AB. The same base, in this case 10, is used throughout the calculation.

Is it physical to take the logarithm of anything with units?

Yes, logarithms always give dimensionless numbers, but no, it’s not physical to take the logarithm of anything with units. Instead, there is always some standard unit. For your example, the standard is the kilometer.

Why are my arguments to a logarithm always dimensionless?

The reason behind this problem is that you’ve not yet simplified the final expression. For example, let’s suppose you get a term $\\ln (f(v))$in your final indefinite integral, where $f(v)$has dimensionsand isn’t dimensionless. This is, as you noted, weird as a logarithm’s arguments should always be dimensionless.

Is the log of 1 km dimensionless or dimensionful?

This shows that the log of 1 km is neither a dimensionless quantity nor a dimensionful one. If it was dimensionless, then it would be expressible without reference to any system of units. If it was dimensionful, then it would change by multiplication when the system of units was changed.

Does log make sense for an object with no units?

Now, implicitly in asserting that log makes sense for objects with units, (and similarly that exp makes sense for objects of units), it is necessary that we already work in a system, that of the graded R -algebra, in which you can add a scalar (an object with no units) to a vector (some object with units).