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Change of Base Formula Calculator

Online change of base calculator: convert log_a b to natural or common logs so a standard calculator can evaluate it.

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log_a b = ln b / ln a
a>0,a≠1
b>0

📖 Tutorial | Change of Base

1. Definition
Change of base: log_a b = log_c b / log_c a (a,b>0,a≠1), usually with c=e or 10.
2. Formula & Notation
log_a bWanted logarithm
ln b, ln aNatural logs
cNew base
3. Properties & Laws
  • log_a b=ln b/ln a=lg b/lg a;
  • Corollary: log_a b=1/log_b a;
  • Chain: log_a b·log_b c=log_a c;
  • c=a restores it.
4. Steps
  1. Enter the base a and number b;
  2. Click "Calculate" (uses ln);
  3. Or click "Load Example".
5. Worked Examples
Example: log₂32=ln32/ln2=5.
6. Common Pitfalls
Numerator ln b, denominator ln a — do not swap;
a,b positive, a≠1;
log_a b and log_b a are reciprocals.

❓ FAQ | Change of Base

Formula?
log_c b/log_c a.
Why?
Calculators have only ln/lg.
Common form?
ln b/ln a.
Corollary?
1/log_b a.
Conditions?
a,b>0,a≠1.
Value changes?
No, exactly equal.