Logarithm Calculator
Find log base b of x, along with ln(x), log₁₀(x) and log₂(x).
Your numbers
Your results
log_b(x)
log_b(x)
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Natural log ln(x)
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log₁₀(x)
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log₂(x)
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Logarithm Calculator by TheCalcTools
How the Logarithm Calculator works
Enter the number x and the base b.
The calculator uses the change-of-base rule: log_b(x) = ln(x) ÷ ln(b).
ln, log₁₀ and log₂ are shown alongside for comparison.
The formula
log_b(x) = y means bʸ = xlog_b(x) = ln(x) ÷ ln(b)
Worked example
Using the default values shown in the calculator:
| Number (x) | 1000 |
|---|---|
| Base (b) | 10 |
- log₁₀(1000) asks: 10 to what power gives 1000?
- Using change of base: ln(1000) ÷ ln(10) = 3.0000.
- ln(1000) = 6.9078 and log₂(1000) = 9.9658.
Result:
| log_b(x) | 3.0000 |
|---|---|
| Natural log ln(x) | 6.9078 |
| log₁₀(x) | 3.0000 |
| log₂(x) | 9.9658 |
Frequently asked questions
What is the logarithm of 1?
It is 0 for any valid base, because any base to the power 0 is 1.
Why must x be positive?
No real power of a positive base gives zero or a negative number, so the real logarithm does not exist there.
Sources and further reading
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This calculator gives estimates for information only. Check important results independently before you rely on them. See our disclaimer.