Math
Change of Base Formula Calculator
Most calculators only have log (base 10) and ln (base e) keys.
The change of base formula rewrites any logarithm using those: log_b(x) = ln(x) / ln(b) = log(x) / log(b). Enter x and the base b to see both routes worked out and an exact answer when one exists.
The number inside the log. Must be positive.
Positive and not equal to 1.
Try an example
Result
log_b(x)
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- Exact form
- —
- Using ln
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- Using log₁₀
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- Check
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The change of base formula
A The exponent you would need to raise a base to in order to get a number. in one The number being raised to a power, or the side a shape's height is measured from. can be rewritten as a ratio of logarithms in any other base. This matters because calculators only offer base 10 and base e, but problems use base 2, 3, 5, or anything else.
Why it works
Let y = log_b(x), so b^y = x. Take ln of both sides: ln(b^y) = ln(x). The power rule pulls y out front: y × ln(b) = ln(x). Divide by ln(b) and you get y = ln(x) / ln(b). The same argument works with log₁₀ or any other base.
Worked example: log base 3 of 81
The result is exactly 4 because 3^4 = 81. When the decimal comes out as a whole number or a simple fraction, the calculator verifies it exactly instead of trusting the rounded division.
Handy consequences
- log_b(a) = 1 / log_a(b): swapping argument and base flips the value.
- log_b(x) and log_c(x) differ only by the constant factor log_b(c).
- log_(1/b)(x) = -log_b(x): a The number you get by flipping a fraction; 3/4 becomes 4/3, and 5 becomes 1/5. base flips the sign.
How to use it
- Enter the argument x (the number inside the log).
- Enter the base b.
- Read the answer and compare the ln route with the log₁₀ route; both give the same value.
- If x is a power of b, the exact whole-number or fraction answer is shown.
How to read the answer
The result is the The small raised number saying how many times to multiply the base by itself. that turns b into x. A whole number means x is an exact power of b. The ln and log₁₀ routes always agree because the ratio of two logs does not depend on which base you measure them in.
Common mistakes and edge cases
- Dividing in the wrong order: it is log(x) / log(b), argument over base.
- Writing log(x / b) instead of log(x) / log(b). Those are different.
- Using a base of 1 or a negative base. Both are undefined.
- Rounding ln values to two decimals before dividing, which can shift the answer in the second decimal place.
Worked examples
Exact power
log₃(81) = 4
log_b(x)
4
Not an exact power
log₅(30)
log_b(x)
2.11328275
Fractional exact
log₄(8) = 3/2 because 4^(3/2) = 8
log_b(x)
1.5
Negative result
log₂(0.125) = -3
log_b(x)
-3
Base less than 1
log₀.₅(8) = -3
log_b(x)
-3
Decimal base
log₁.₀₅(2): doubling periods at 5%
log_b(x)
14.20669908
Invalid base
Base 1 is not allowed
log_b(x)
Error
Frequently asked questions
What is the change of base formula?+
log_b(x) = log_c(x) / log_c(b). Any base c works; on a calculator you use c = 10 (the log key) or c = e (the ln key).
Should I use ln or log for change of base?+
Either. Both give the same answer because the ratio cancels the base. Use whichever key your calculator has; this page shows both.
How do I find log base 3 of 81?+
ln(81) / ln(3) = 4.3944 / 1.0986 = 4. Or notice 3^4 = 81 directly.
Why can't the base be 1?+
1 to any power is 1, so log base 1 could never produce numbers other than 1, and ln(1) = 0 would put a zero in the denominator.
Can I use the change of base formula backward?+
Yes. log(x) / log(b) can be collapsed back to log_b(x). That is useful for simplifying ratios of logs.
Does the formula work for negative or zero arguments?+
No. Logarithms of zero or negative numbers are undefined in real numbers, whatever the base.
About this calculator
- Written by
- mathcheck editorial team
- Last reviewed
- September 4, 2026
Method
- Uses the values entered by the user and stable formulas documented on the page.
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Last updated: September 4, 2026