Logarithmic gain has the dimension length per time. It is defined as the logarithm of a measurement in a given unit per a base measurement.

Maple knows the units of logarithmic gain listed in the following table.

An asterisk ( * ) indicates the default context, an at sign (@) indicates an abbreviation, and under the prefixes column, SI indicates that the unit takes all SI prefixes, IEC indicates that the unit takes IEC prefixes, and SI+ and SI- indicate that the unit takes only positive and negative SI prefixes, respectively.  Refer to a unit in the Units package by indexing the name or symbol with the context, for example, mach[standard] or mph[standard]; or, if the context is indicated as the default, by using only the unit name or symbol, for example, mach or mph.

In the Standard and Natural Units environments, you can take the logarithm of a quantity that has a unit of the form $\frac{x}{x\left(\mathrm{base}\right)}$, for any valid unit $x$ (see Units,annotations for an explanation of what the symbol in parentheses means), and obtain a result with units in the dimension of logarithmic gain.  Similarly, you can raise a unit-free quantity to a quantity with units in the dimension of logarithmic gain, and obtain a result with the dimension $\frac{\mathrm{length}\left(\mathrm{base}\right)}{\mathrm{length}}$. (This corresponds to the case above for $x$ being a unit of dimension $\frac{1}{\mathrm{length}}$.) As a special case, you can apply the exponential function to a quantity with units in the dimension of logarithmic gain to the same effect. See the Units[Standard][ln] and Units[Standard][exp] help pages for more details.

None of these computations are legal in the Simple Units environment. In the default Units environment, they are legal if you call the combine/units command in standard or natural mode, but not in simple mode.

The units of logarithmic_gain are defined as follows.

A bel is defined as the logarithm base 10 of a power over a base power.  Because power is proportional to the square of the voltage, the conversion factor from nepers to bels is $\frac{2}{\ln \left(10\right)}$.

$\mathrm{convert}\left(13.2\,'\mathrm{units}'\,'\mathrm{Np}'\,'\mathrm{dB}'\right)$

$114.6537432$

$\mathrm{convert}\left('\mathrm{amperes}'\,'\mathrm{dimensions}'\,'\mathrm{energy}'\right)$

$\frac{1}{\mathrm{length}}$

$\mathrm{convert}\left('\mathrm{watts}'\,'\mathrm{dimensions}'\,'\mathrm{energy}'\right)$

$\frac{1}{{\mathrm{length}}^2}$

$\mathrm{convert}\left('\mathrm{pascals}'\,'\mathrm{dimensions}'\,'\mathrm{energy}'\right)$

$\frac{1}{{\mathrm{length}}^4}$

$\mathrm{with}\left(\mathrm{Units}\left[\mathrm{Standard}\right]\right)\:$

To enter a unit in 2-D Math input, select the unit from the appropriate Units palette. If the unit you want is not there, select $\mathrm{unit}$ and then enter the unit.

When you edit a unit, double brackets appear around it.

$\ln \left(3.2\mathrm{Unit}\left(\frac{'W'}{'W\left(\mathrm{base}\right)'}\right)\right)$

$0.5815754050⟦\mathrm{Np}⟧$

$\mathrm{convert}\left(\,'\mathrm{units}'\,'\mathrm{dB}'\right)$

$5.051499784⟦\mathrm{dB}⟧$