# Exponents, Roots, and Logarithms

`D00-F01-A05` · Financial Mathematics, Statistics, and Data Foundations → Mathematical Language and Quantitative Reasoning · archetype `record-transform` · difficulty 1/5 · verification **verified**

Full page: https://docs.thefintechbuilder.com/foundations/mathematical-language-and-quantitative-reasoning/exponents-roots-and-logarithms/
Agent skill: `npx skills add IslamBaraka90/Fintech-Algorithms-Library` — https://docs.thefintechbuilder.com/guides/agent-skill/

## Install and import

```bash
npm install fintech-algorithms
```

```ts
import { exponentsRootsAndLogarithms } from "fintech-algorithms/foundations/mathematical-language-and-quantitative-reasoning/exponents-roots-and-logarithms";
```

## Signature

```ts
exponentsRootsAndLogarithms(input)
```

Applies the three operations that share one base: raising it to an exponent, taking its square root, and taking its natural logarithm.

## Parameters

| Name | Type | Required | Notes |
| --- | --- | --- | --- |
| `input` | `D00Input` | yes | A plain object. Every F01 topic first reads `values`, which must be a non-empty array of finite numbers, and validates it before any per-topic branch runs. This topic then reads `base` and `exponent`, both coerced with `Number`; `values` is demanded but not used by this calculation. |

## Returns

`{ power: number; squareRoot: number; naturalLog: number }`

`power` is `base` raised to `exponent`, `squareRoot` is the square root of `base`, and `naturalLog` is its natural logarithm.

## Errors

- When input is not a plain object — throws TypeError
- When values is missing, empty, or not an array — throws RangeError
- When values contains a value that is not a finite number — throws RangeError
- When base is zero or negative — throws RangeError

## Complexity

Time `O(n)`, space `O(n)`.

## Worked example

This is the worked example published in the article, replayed by the test suite on every run. The output cannot drift.

### Input

`input`:

```json
{
  "values": [1, 2, 3, 4, 5],
  "startingCash": 1000,
  "inflows": 250,
  "outflows": 120,
  "slope": 2,
  "intercept": 1,
  "part": 25,
  "whole": 200,
  "elapsed": 5,
  "old": 4.5,
  "new": 4.75,
  "base": 16,
  "exponent": 0.5,
  "a": 2
}
```

Showing 14 of 28 fields.

### Call

```ts
exponentsRootsAndLogarithms(input)
```

### Returns

object with 2 fields: power, squareRoot

```json
{
  "power": 4,
  "squareRoot": 4
}
```

## Verification and provenance

Tier: **verified** (via input-expected).

The worked example below is the figure published in this algorithm's article, replayed and asserted by the test suite on every build. The arithmetic cannot drift without the build failing.

Both tiers guarantee the signature. Full explanation: https://docs.thefintechbuilder.com/guides/verification/

Generated from the docs.json payload shipped inside fintech-algorithms@0.13.0.
The signature and parameter list are checked against the compiled implementation at build time,
so a description that contradicts the code fails the build rather than reaching this file.

## Links

- Article (how it works, step by step): https://thefintechbuilder.com/foundations/mathematical-language-and-quantitative-reasoning/exponents-roots-and-logarithms/
- Implementation source: https://github.com/IslamBaraka90/Fintech-Algorithms-Library/blob/main/src/foundations/mathematical-language-and-quantitative-reasoning/exponents-roots-and-logarithms/impl.ts
- Package on npm: https://www.npmjs.com/package/fintech-algorithms
- Domain index for agents: https://docs.thefintechbuilder.com/foundations/llms.txt