Log Calculator
Solve log_b(x) = y for any base — including natural log and common log — or work backward to find x or the base itself. Full antilog, log rules, and comparison tables included below.
2
LOG_B(X)
4.6052
LN(X)
6.6439
LOG₂(X)
Solve log_b(x) = y
⚡ Choose which value to solve for above, then fill in the other two. Formula: log_b(x) = ln(x) / ln(b).
Result · log10(100)
2
10^2 = 100
100
Antilog (b^y)
✓ Verified
Check b^y = x
2
LOG_B(X)
2
LOG₁₀(X)
4.6052
LN(X)
6.6439
LOG₂(X)
Base Comparison Table
log(x) for x = 100 across the three most common bases.
| Base | Notation | log(x) |
|---|---|---|
| 2 | Binary Log (log₂) | 6.6439 |
| 10 | Common Log (log₁₀) | 2 |
| 2.7183 | Natural Log (ln) | 4.6052 |
All Log Bases — Same x Value
Common Log (log₁₀)
2
log₁₀(100), base 10
Natural Log (ln)
4.6052
ln(100), base e ≈ 2.718
Binary Log (log₂)
6.6439
log₂(100), base 2
Custom Log (log_b)
2
log_10(100), base 10
Antilogarithm (Inverse Log)
Given log_base(x) = y, the antilog finds x back from y — i.e. base^y. Here it’s applied to your current result, y = 2.
Antilog base 10 of y
100
10^y
Antilog base e of y
7.3891
e^y
Antilog base 2 of y
4
2^y
Antilog base b of y
100
10^y
√x (square root)
10
x^(1/2)
∛x (cube root)
4.6416
x^(1/3)
Logarithm Curves — log₂, ln, log₁₀
Logarithm Rules Reference
Live examples below use your current base (10) and argument (100).
Product Rule
log_b(xy) = log_b(x) + log_b(y)
log_10(100×2) = log_10(100) + log_10(2) = 2.301
Quotient Rule
log_b(x/y) = log_b(x) − log_b(y)
log_10(100/2) = log_10(100) − log_10(2) = 1.699
Power Rule
log_b(xⁿ) = n · log_b(x)
log_10(100²) = 2 × log_10(100) = 4
Change of Base
log_b(x) = log_k(x) / log_k(b)
log_10(100) = log₁₀(100)/log₁₀(10) = 2/1 = 2
Identity Rule
log_b(b) = 1, log_b(1) = 0
log_10(10) = 1, log_10(1) = 0
Inverse Rule
b^(log_b(x)) = x
10^(log_10(100)) = 10^2 ≈ 100
Reciprocal Base
log_b(x) = 1 / log_x(b)
log_10(100) = 1 / log_100(10) = 2
Natural Log Form
log_b(x) = ln(x) / ln(b)
ln(100) / ln(10) = 4.6052 / 2.3026 = 2
Understanding Logarithms
A logarithm is the inverse of exponentiation. Where an exponent starts with a base and a power to find a result — the job of our exponent calculator — a logarithm starts with a base and a result and works backward to recover the power: if bʸ = x, then log_b(x) = y. In plain terms, log_b(x) asks “to what power must I raise b to get x?” This calculator solves that question for any of the three variables, not just the log value itself.
Key Logarithm Formulas
Definition: log_b(x) = y ↔ bʸ = xCommon log: log(x) = log₁₀(x)Natural log: ln(x) = log_e(x), where e ≈ 2.71828Change of base: log_b(x) = ln(x) / ln(b) = log(x) / log(b)Antilogarithm: x = bʸ (find x from y) | b = x^(1/y) (find base from x and y)Where logarithms show up: pH in acid/base chemistry, the decibel scale for sound, the Richter scale for earthquakes, information entropy in computer science, compound-interest and doubling-time problems in finance, and the time complexity of many search and sort algorithms (Big-O notation). Natural log in particular underpins calculus, since the derivative of ln(x) is simply 1/x.
— FAQ