Logarithm Calculator
Calculate log base b of x with change-of-base formula, exponential form, and step-by-step solution.
Logarithm
Updates as you typeMode
What do you want to solve for? ?
Base
Common bases ?
Custom base (b) ?
—
Inputs
Number (x) ?
—
log2(
)
Exponent (y) ?
—
Result value (y) ?
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Quick values (click to fill)
Common x
Display (optional)
Decimal precision ?
The curve y = log2(x)
x range: 0,1 – 16
Zoom x-axis
Formula
logb(x)
=
y
⇔
by
=
x
Change of base: logb(x) = ln(x) / ln(b) = log10(x) / log10(b)
log2(10) = ln(10) / ln(2) ≈ 2,3026 / 0,6931 ≈ 3,3219
Product rule: logb(a × c) = logb(a) + logb(c)
log2(8 × 4) = log2(8) + log2(4) = 3 + 2 = 5
Quotient rule: logb(a / c) = logb(a) − logb(c)
log10(100 / 10) = log10(100) − log10(10) = 2 − 1 = 1
Power rule: logb(an) = n × logb(a)
log2(83) = 3 × log2(8) = 3 × 3 = 9
Inverse: blogb(x) = x, logb(bx) = x
10log10(5) = 5, log10(105) = 5
Reciprocal bases: log1/b(x) = −logb(x)
log0,5(8) = −log2(8) = −3
- b
- Base — any positive number except 1. Common choices: 2 (binary), e ≈ 2.71828 (natural), 10 (common).
- x
- Argument — strictly positive. log of 0 is −∞; log of a negative number is undefined in the reals.
- logb(1)
- Always 0 for any valid base (b⁰ = 1).
- logb(b)
- Always 1 (b¹ = b).
- ln(x)
- Shorthand for logₑ(x) — the natural logarithm.
Worked example — your numbers
- Base: —
- Argument: —
- Setup: —
- Change of base: —
- ln ratio: —
- Result: —
- Check: —
Logarithms turn multiplication into addition — that is why log-scale axes compress huge ranges into something readable. They are the inverse of exponentials: whatever you can say with bʸ = x, you can rephrase as log_b(x) = y.