Common Logarithm Calculator
Compute log₁₀(x), invert to find x, or change base. Live formula playground with reference values.
Input
Updates as you typeMode
What do you want to solve for? ?
Argument
Number x ?
—
Target
Value of log₁₀(x) = y ?
—
Base
Target base b ?
—
y = log₁₀(x)
Current point markedx = — · log₁₀(x) = —
y = log₁₀(x)
Current x
Common values
Click a row to load it| x | log₁₀(x) | ln(x) | Note |
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Identity playground
Verify log rules with your numbersValue a
Value b
Product rule: log(a·b)
—
= log(a) + log(b)
—
Quotient rule: log(a/b)
—
= log(a) − log(b)
—
Power rule: log(aᵇ)
—
= b · log(a)
—
Where log₁₀ shows up
Tap a card to load its xFormula
log10(x)
=
log base 10 of x
,
10y = x ⇔ y = log10(x)
- x
- Argument of the logarithm; must be strictly > 0
- Base 10
- The "common" base — aligns with decimal place value
- log10(1) = 0
- Any logarithm of 1 is zero
- log10(10) = 1
- Logarithm of the base equals 1
- log10(10n) = n
- The log of a power of 10 is its exponent
Product: log10(a · b) = log10(a) + log10(b)
Quotient: log10(a / b) = log10(a) − log10(b)
Power: log10(aᵏ) = k · log10(a)
Reciprocal: log10(1/a) = −log10(a)
Derivative: d/dx log10(x) = 1 / (x · ln(10)) ≈ 0.4343 / x
Integral: ∫ log10(x) dx = x · log10(x) − x / ln(10) + C
Series: log10(1 + u) = (u − u²/2 + u³/3 − …) / ln(10)
Limit: limx→0⁺ log10(x) = −∞, limx→∞ log10(x) = ∞
Inverse: log10(10x) = x and 10log10(x) = x
Change of base: logb(x) = log10(x) / log10(b)
Bridge to ln: log10(x) = ln(x) / ln(10) ≈ 0.4343 · ln(x)
Order of magnitude: ⌊log₁₀(x)⌋ gives the exponent in scientific notation
Worked example — your numbers
- Input: x = —
- log₁₀(x) = the exponent y such that 10ʸ = x
- log10(—) = —
- Exponential form: —
- Bridge to ln: log10(x) = ln(x) / ln(10) ≈ —
Common logs scale quantities that span many orders of magnitude — acidity (pH), loudness (dB), earthquake energy (Richter), star brightness, and anything reported in "decades." The base-10 choice also makes ⌊log₁₀(x)⌋ + 1 the number of digits in x, which is why it shows up in coding, combinatorics, and information-theoretic bounds.