Log Calculator: Base 10, ln, log₂, Custom

Log calculator: base 10, ln, log₂, and custom base

Evaluate log_b(x) for base 10, e, 2, or a custom base. Inverse proof on every result; change-of-base steps in Custom mode. Positive x and valid base only; runs locally.

What this calculator does

The widget returns y = log_b(x): the exponent that raises base b to x. Pick the Mode dropdown for base 10, e (ln), 2, or Custom base (typed b and x). The dark results card labels the active base in log_b(x) (10, e, 2, or your typed b). A green Inverse proof prints b^y = x; Custom mode also shows a blue Change of base line with ln(x)/ln(b). Requires x > 0 and base > 0 with base ≠ 1. No graphing, symbolic log laws, or complex logs.

Definition:

$\log_b(x)=y\ \Longleftrightarrow\ b^y=x$
Evaluation:

$\log_b(x)=\frac{\ln(x)}{\ln(b)}$

How the math works

Every mode uses ln(x)/ln(b) internally; Custom mode prints that ratio with your numbers. Secondary check (not a form default): log₅(125) = 3 because 5³ = 125. Product, quotient, and power rules for logs are not expanded on this page.

Limits

Logarithms are undefined for x ≤ 0 in real arithmetic. Base must be positive and not 1. The widget shows an amber domain message instead of a numeric result when inputs break those rules. Display trims trailing zeros after up to 10 decimal places. Natural log with x = 2.718 is slightly below 1 because the field uses a decimal approximation of e, not the full constant; type a higher-precision x if you need ln(x) closer to exactly 1.

Log Calculator FAQ

What result does each mode show at its defaults?

Common log, x = 100 → 2. Natural log, x = 2.718 → 0.9998963157 (2.718 is a coarse stand-in for e). Binary log, x = 8 → 3. Custom base, b = 3, x = 27 → 3. Each mode keeps its own x (and b) when you switch back.

What is the change of base formula on this page?

Custom base evaluates

\log_b(x)=\frac{\ln(x)}{\ln(b)}

and prints ln(x) and ln(b) with your numbers in the blue panel. Other modes use the same math internally; only Custom shows the substitution.

What does the inverse proof box check?

After a valid result, the green panel shows b^y = x using your base, exponent y from the results card, and input x. The fine print compares b^y to a floating-point check (on the common-log default, 10² ≈ 100.000000).

Why does the widget show a domain error?

x must be greater than 0. Custom base must be positive and not equal to 1. Messages include Number must be greater than 0 and Base must be positive and not equal to 1. Invalid text blocks the dark results card until inputs are valid.

What is the difference between log, ln, and log₂ here?

Same operation, different base from the Mode dropdown: 10, e (ln), 2, or a typed custom base. The readout labels log_b(x) using the active base.

How many decimal places does the results card show?

Up to 10 decimal places; trailing zeros are trimmed. Integers like log₁₀(100) = 2 print without a decimal part.

What does the results card show before inputs are valid?

The placeholder reads Enter a positive number with a note that custom base needs a positive base ≠ 1. Domain errors replace the numeric readout with an amber alert under the inputs until x and (if custom) b are valid.

When does the change of base panel appear?

Only in Custom base mode after b and x pass domain checks. Common, natural, and binary modes still use ln(x)/ln(b) internally but hide the blue substitution panel.

Do inputs persist when I switch modes?

Yes. Each mode keeps its own x field (and custom keeps b = 3, x = 27). Switching back to Common log restores x = 100 without overwriting other modes.

Can this solve log equations symbolically?

No. It evaluates one numeric log per run. It does not graph curves, expand log laws on an expression, or accept complex arguments.

Log Calculator: Base 10, ln, log₂, and Custom

Logarithm

Inverse proof

How to use this calculator

Reading your logarithm result

Example: Common log (default) → log₁₀(100) = 2

Example: Custom base → log₃(27) = 3