log10

Math

Computes the base-10 logarithm of a positive 18-decimal fixed-point input via the change-of-base identity log₁₀(x) = ln(x) / ln(10).

Gas

391

Max rel. error

1.4e-14

Signature

solidity

function log10(uint256 x) internal pure returns (int256 y)

Parameters

Name	Type	Description
x	uint256	Input in 18-decimal fixed-point format (1e18 = 1.0). Must satisfy x > 0.

Returns

Name	Type	Description
y	int256	Result log₁₀(x) in 18-decimal fixed-point format. Signed — returns negative values for x < 1 (i.e. x < 1e18).

Bounds

Bound	Value
Input domain	Full `uint256` domain except `x == 0` (see Errors). Inherits `ln`'s unbounded acceptance — no named upper or lower bound constants.

Behavior

Implemented as ln(x) · 1e18 / 2.302585…e18 — a single multiply-and-divide on top of ln.
Reverts with LnLowerBoundError() on x == 0, inherited from ln.
Returns 0 when x == 1e18 (i.e. log₁₀(1) = 0); negative when x < 1e18; positive when x > 1e18.
Precision inherits from ln; the change-of-base division adds a negligible ulp of rounding.
Pure internal function; no external calls or storage.

How it works

The change-of-base identity converts any logarithm into a single division by a constant:

log₁₀(x) = ln(x) / ln(10)

DeFiMath stores ln(10) as the 18-decimal constant 2302585092994045684 (precision ~5e-19 vs. the true value). The implementation is one line:

y = ln(x) * 1e18 / 2302585092994045684;

The multiplication by 1e18 before dividing keeps the result in 18-decimal fixed-point format. Total cost is one ln call (~375 gas) plus a 16-gas mul+div, hence the ~391 gas total.

Because the implementation is a thin wrapper over ln, every guarantee ln provides — domain (x > 0), precision, sign behavior — flows through directly. The only added error term is the rounding of ln(10) to 18 decimals, which is dwarfed by ln's own ~1e-14 error.

Errors

Error	Trigger
LnLowerBoundError	`x == 0` (inherited via the internal call to `ln`)

Example

solidity

import "defimath-lib/contracts/math/Math.sol";

uint256 x   = 1000e18;               // x = 1000
int256  l10 = DeFiMath.log10(x);     // l10 ≈ 3e18  (= log₁₀(1000))

uint256 y   = 0.1e18;                // y = 0.1
int256  ly  = DeFiMath.log10(y);     // ly  ≈ −1e18 (= log₁₀(0.1))

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