Free · Exponents and logarithms

Negative Logarithm with a Chosen Base

Calculate minus the logarithm of a positive number using a positive base other than one.

Formula & worked example
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How to use this calculator

Enter the known values in the units shown. Results update as you type.
Where results are editable, change one to solve backwards. Lock a value to hold it fixed.
Use the worked example to check the method. Reset restores the starting fields.

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Formula and method

For x = 0.01 and base 10, −log₁₀(x) = 2. For x = 100 with the same base, the result is −2.

Negative logarithm = −log_b(x) = ln(1/x) ÷ ln(b)

Worked example

Enter these known values and leave the other values blank.

Positive number x: .01
Logarithm base b: 10

Negative logarithm −log_b(x): 2

Assumptions and limitations

The number and base must be positive, and the base must not equal one.
The default base is 10. Negative logarithm means the negative of a logarithm, not the logarithm of a negative number.
A result of zero occurs at x = 1 for every allowed base, so that pair cannot identify a unique base.

Common questions

Can the answer be negative?

Yes. With base 10, any input above 1 produces a negative result.

Can I use a base smaller than 1?

Yes, if it is positive. For example, −log₀.₅(0.25) is −2.

References

The calculation equations, inverse formulas, units, and input rules were imported from this source. Bookify provides the interface and equation solver.

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