Logarithm Calculator
Calculate log base b, log10, natural log (ln) and log2 with change-of-base formula and antilog.
Must be > 0
Must be > 0, ≠ 1
log2(8) =
3
Verification: 2^3 = 8 ✓
Change of Base Formula
log2(8) = ln(8) / ln(2)
= 2.079441542 / 0.6931471806
= 3
Standard Logarithms of x = 8
Common log
log₁₀(x)
0.903089987
10
Natural log
ln(x)
2.079441542
e
Binary log
log₂(x)
3
2
Antilogarithm (Inverse)
10^0.903089987= 8
e^2.079441542= 8
2^3= 8
Logarithm Rules
Product: log_b(xy) = log_b(x) + log_b(y)
Quotient: log_b(x/y) = log_b(x) − log_b(y)
Power: log_b(xⁿ) = n · log_b(x)
Identity: log_b(b) = 1
Zero: log_b(1) = 0
Change of Base: log_b(x) = ln(x) / ln(b)
LogarithmNatural LogLog Base 2Exponential
Calculate logarithms in any base using the change-of-base formula. Also shows log₁₀, natural log (ln) and log₂ with antilogarithm verification.
Key Features
Custom Base
Computes log_b(x) for any positive base b ≠ 1 using ln(x)/ln(b).
Standard Logs
Shows common log (log₁₀), natural log (ln), and binary log (log₂) for the same x.
Change of Base
Displays the full change-of-base formula with substituted numerical values.
Antilogarithm
Verifies each result: 10^result, e^result and 2^result should equal x.
Log Rules Reference
Built-in quick-reference card covering product, quotient, power, identity, zero and change-of-base rules.
Domain Validation
Clearly explains errors for invalid inputs: x ≤ 0, b ≤ 0, or b = 1.
How to Use
- 1Enter x (the argument) and b (the base).
- 2The result log_b(x) and all standard logarithms appear instantly.
- 3Check the change-of-base formula section for step-by-step details.