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# Logarithm of Probability Density Function
> [Logistic][logistic-distribution] distribution logarithm of [probability density function (PDF)][pdf].
<section class="intro">
The [probability density function][pdf] (PDF) for a [logistic][logistic-distribution] random variable is
<!-- <equation class="equation" label="eq:logistic_pdf" align="center" raw="f(x; \mu,s) = \frac{e^{-\frac{x-\mu}{s}}} {s\left(1+e^{-\frac{x-\mu}{s}}\right)^2}" alt="Probability density function (PDF) for a logistic distribution."> -->
<div class="equation" align="center" data-raw-text="f(x; \mu,s) = \frac{e^{-\frac{x-\mu}{s}}} {s\left(1+e^{-\frac{x-\mu}{s}}\right)^2}" data-equation="eq:logistic_pdf">
<img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@591cf9d5c3a0cd3c1ceec961e5c49d73a68374cb/lib/node_modules/@stdlib/stats/base/dists/logistic/logpdf/docs/img/equation_logistic_pdf.svg" alt="Probability density function (PDF) for a logistic distribution.">
<br>
</div>
<!-- </equation> -->
where `mu` is the location parameter and `s` is the scale parameter.
</section>
<!-- /.intro -->
<section class="usage">
## Usage
```javascript
var logpdf = require( '@stdlib/stats/base/dists/logistic/logpdf' );
```
#### logpdf( x, mu, s )
Evaluates the logarithm of the [probability density function][pdf] (PDF) for a [logistic][logistic-distribution] distribution with parameters `mu` (location parameter) and `s` (scale parameter).
```javascript
var y = logpdf( 2.0, 0.0, 1.0 );
// returns ~-2.254
y = logpdf( -1.0, 4.0, 4.0 );
// returns ~-3.14
```
If provided `NaN` as any argument, the function returns `NaN`.
```javascript
var y = logpdf( NaN, 0.0, 1.0 );
// returns NaN
y = logpdf( 0.0, NaN, 1.0 );
// returns NaN
y = logpdf( 0.0, 0.0, NaN );
// returns NaN
```
If provided `s < 0`, the function returns `NaN`.
```javascript
var y = logpdf( 2.0, 0.0, -1.0 );
// returns NaN
```
If provided `s = 0`, the function evaluates the logarithm of the [PDF][pdf] of a [degenerate distribution][degenerate-distribution] centered at `mu`.
```javascript
var y = logpdf( 2.0, 8.0, 0.0 );
// returns -Infinity
y = logpdf( 8.0, 8.0, 0.0 );
// returns Infinity
```
#### logpdf.factory( mu, s )
Returns a function for evaluating the logarithm of the [probability density function][pdf] (PDF) of a [logistic][logistic-distribution] distribution with parameters `mu` (location parameter) and `s` (scale parameter).
```javascript
var mylogpdf = logpdf.factory( 10.0, 2.0 );
var y = mylogpdf( 10.0 );
// returns ~-2.079
y = mylogpdf( 5.0 );
// returns ~-3.351
```
</section>
<!-- /.usage -->
<section class="notes">
## Notes
- In virtually all cases, using the `logpdf` or `logcdf` functions is preferable to manually computing the logarithm of the `pdf` or `cdf`, respectively, since the latter is prone to overflow and underflow.
</section>
<!-- /.notes -->
<section class="examples">
## Examples
<!-- eslint no-undef: "error" -->
```javascript
var randu = require( '@stdlib/random/base/randu' );
var logpdf = require( '@stdlib/stats/base/dists/logistic/logpdf' );
var mu;
var s;
var x;
var y;
var i;
for ( i = 0; i < 10; i++ ) {
x = randu() * 10.0;
mu = randu() * 10.0;
s = randu() * 10.0;
y = logpdf( x, mu, s );
console.log( 'x: %d, µ: %d, s: %d, ln(f(x;µ,s)): %d', x, mu, s, y );
}
```
</section>
<!-- /.examples -->
<section class="links">
[logistic-distribution]: https://en.wikipedia.org/wiki/Logistic_distribution
[pdf]: https://en.wikipedia.org/wiki/Probability_density_function
[degenerate-distribution]: https://en.wikipedia.org/wiki/Degenerate_distribution
</section>
<!-- /.links -->
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