Logarithm Change Of Base
Calculator
Results
- Logarithm in the starting base
- 2.999999
- Logarithm in the target base
- 9.965784
- Natural logarithm
- 6.907755
- Base-10 logarithm
- 3
Results
| Logarithm in the starting base | 2.999999 |
| Logarithm in the target base | 9.965784 |
| Natural logarithm | 6.907755 |
| Base-10 logarithm | 3 |
formula-map diagram
- Logarithm in the starting base
- 2.999999
- Logarithm in the target base
- 9.965784
- Natural logarithm
- 6.907755
- Base-10 logarithm
- 3
Formula map
Formula
log_b(x) = ln(x) ÷ ln(b)= 3
Note
Simplified model: these are exact results of the stated idealised formulas, assuming perfect shapes and exact inputs. Real measurements carry error, and the ellipse perimeter is a Ramanujan approximation. Verify independently before relying on these figures.
More in Mathematics
See all →Frequently asked questions
Why would I need to change the base of a logarithm?+
Most calculators and software only compute logarithms in base 10 or base e (natural log), so to find a logarithm in another base — like base 2 for information theory — you need to convert it using a known base.
What is the change-of-base formula?+
log_b(x) = log_k(x) / log_k(b), where k is any base you can compute directly (usually 10 or e). You compute the log of x and the log of b in the same known base, then divide.
What inputs does this calculator need?+
The number you're taking the log of (x), and the base you want the logarithm expressed in (b). It computes log_k(x) and log_k(b) internally and divides them.
Why must the base and the argument both be positive?+
Logarithms are only defined for positive arguments, and a valid base must be positive and not equal to 1 — otherwise the underlying exponential relationship breaks down and no real answer exists.
Does the choice of k (10 or e) affect the final answer?+
No, the change-of-base formula gives the exact same result regardless of which valid base k you compute through — it's purely a computational convenience, not a change in the mathematical answer.