# e^t, where t is time. Both the exponential function and its argument must be dimensionless. ?

Clearly the function has a "t" as a unit of time. How can this equation be valid. Can anyone help me understand this paradox?

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• Vaman
Lv 7
1 month ago

This equation is not valid. The term in the exponent shoud be dimensionless. You need e^(wt) or e^(t/T) type.

• Dixon
Lv 7
1 month ago

You are correct, raising to a power means the power must be just a number. Typically you will see t/τ where τ is some kind of "time constant" in the same units as t. So if t is in years and τ = 100 years, this would effectively give you e^(number of centuries)

• 1 month ago

Yes, you're right.  Technically, it's not valid.  Generally, in physical sciences you'll find that a formula involving exponential growth in time has coefficients with units that cancel out or apply dimensionality as needed;  something of the form:

f(t) = A e^(bt)

The units of b will be such that bt is dimensionless, while the dimensions of A will be those expected for the function result.  The same is true for powers with a variable exponent or with a variable base and an irrational exponent, as well as for logarithms, trig functions and other special functions.

Something similar also happens in polynomials, where an equation like

f(t) = at^2 + bt + c

....had different dimensonality on each constant so that the terms all have the dimensionality required for the function result.

In math textbooks, though, dimensionality is often ignored--even in so-called "real world" problems.