To calculate natural logarithms, enter 'e' above.

Negative Log (base 10)
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This calculator computes the negative logarithm (-log) of a number for a given base.

This tool finds the negative logarithm of a number for any base you choose. You can use base 10, base 2, base e (natural log), or any base > 0 and ≠ 1.

Formula

Negative log (base b) of x:

−log_b(x)

Change of base form:

−log_b(x) = − ln(x) / ln(b)

For natural log, set base to e and it becomes −ln(x).

Your example

Number: 76

Base of logarithm: 106

Negative Log (base 106): −0.9287

Work (rounded): − ln(76) / ln(106) ≈ −0.9287

How to use the calculator

  1. Enter the number (x). It must be > 0.
  2. Enter the base (b). It must be > 0 and not equal to 1. For natural logs, type e.
  3. Click calculate. The result shows −log_b(x).
  4. Choose your rounding if needed. Two to four decimals are common.

Notes and tips

  • If x > 1 and b > 1, then log_b(x) is positive, so the negative log is negative.
  • If 0 < x < 1 and b > 1, then log_b(x) is negative, so the negative log is positive.
  • Use the same base rules for probability work. Many people use −log_{10}(p) or −ln(p).
  • Inputs of x ≤ 0 are not valid. Base b ≤ 0 or b = 1 is not valid.

Extra examples

Example 1: Natural log

−ln(0.2) ≈ 1.6094 (positive because 0.2 < 1)

Example 2: Base 10

−log10(50) ≈ −1.6990

Example 3: Base 2

−log2(0.5) = 1

FAQ

What is a negative log?

It is the value of −log_b(x), the negative of the logarithm of x to base b.

When is the negative log positive?

When 0 < x < 1 (with b > 1), the negative log is positive.

When is the negative log negative?

When x > 1 (with b > 1), the negative log is negative.

Can I use base e?

Yes. Enter e for the base to compute −ln(x).

Is there a change-of-base rule?

Yes. −log_b(x) = − ln(x) / ln(b).

What are valid inputs?

x > 0, b > 0, and b ≠ 1.

Why is this used in statistics?

It turns small probabilities into larger, easier numbers and helps compare likelihoods.

How many decimals should I keep?

Two to four decimals are common. Use more if you need precision.

How do I read the sign?

Positive means the original x was between 0 and 1. Negative means x was greater than 1.

What if I type base 1 or a negative base?

That is invalid for standard logs. Choose a base > 0 and not equal to 1.

Does scaling the base change the meaning?

It changes the numeric value, but not the order when comparing different x values with the same base.

Can I convert between bases later?

Yes. Use the change-of-base rule to move from one base to another.