Logarithm Calculator

Logarithm Calculator — quick guide

1) Core idea

Logarithms are the inverse of exponentials: if b^x = y then log_b(y) = x. They turn multiplication into addition and powers into products.

2) How this tool works

• Log base b: log_b(x) computed via change of base ln(x)/ln(b).
• ln(x) and log10(x): direct helpers for common bases e and 10.
• Antilog: returns b^x for the chosen base.
• Change of base: log_new(x) = log_old(x) / log_old(new).

3) Sanity checks

• Domain: x > 0; base b > 0 and b ≠ 1.
• ln(1) = 0; log10(1000) = 3; log₂(8) = 3.
• log_b(b) = 1; log_b(1) = 0.

4) Shortcuts that help

• Use ln for calculus; log10 for orders of magnitude.
• Exploit properties: product, quotient, and power rules.
• For base-2, use ln(x)/ln(2) or log10(x)/log10(2).

5) Common pitfalls

• Feeding non-positive x or invalid base values.
• Mixing bases without the change-of-base rule.
• Rounding too early—keep full precision, round only for display.

6) Micro-examples

• ln(e^2) = 2
• log10(0.001) = -3
• log₂(32) = 5

7) Mini-FAQ

• Can base be 1? No; logarithm base must be > 0 and ≠ 1.
• What about negative x? Not in the reals; logs are undefined for x ≤ 0.
• Precision? This tool uses JavaScript’s double precision; results are rounded for display only.

8) Action tip

Switch bases with the change-of-base panel, then compare ln vs log10 to build intuition across contexts.