# Negative Log Calculator

Welcome to the Omni's negative log calculator. Using this calculator, you can find the **negative logarithm** of any number with any chosen base. For details on logarithms and how to find the negative log of a number, read the description given below.

## Why do we need to learn about logarithms?

Do you know how many 2's you have to multiply together to get 8?

The answer is easy; `2`

, i.e., you have to multiply three 2's together to get 8.^{3} = 8

But what if we ask you to calculate the number of 7's you need to multiply to get 5,764,801? Not so easy anymore, right?

Thanks to the Scottish mathematician **John Napier**, who invented logarithms as a calculation tool in the 16th century, we can handle numerical expressions that involve multiplications or divisions of large numbers.

**Logarithms are extensively used in chemistry, physics, mathematics, and engineering problems to solve complex calculations with ease.**

Before going any further, let should first try to understand what a logarithm is!

## What is a logarithm?

For any positive real number *a*, and any rational number *n*, let `a`

where ^{n} = b*b* is also a real number.

We can say that the *n-th* power of base *a* is *b* or that we need to multiply *a* by itself *n* times to get *b*. We can also say that the logarithm of *b* to base *a* is *n* and express this mathematically as:

`log`

_{a}(b) = n

To understand this with an example, let us come back to our first problem.

We know that `2`

,
i.e., ^{3} = 8 `log`

. Hence, we can say that the logarithm of 8 to base 2 is 3._{2}(8) = 3

You must have realized by now that 3 is the exponent of 2. Therefore, **while calculating the logarithm of a number, we are simply trying to determine the exponent to which the base must be raised to get that number**.

To find the answer to our second problem, you can use our log calculator. You can also check our antilog calculator to find the antilog of any number.

## How to calculate negative logarithms?

To calculate the negative logarithm of a number, we need to determine how many times we should divide 1 by the base to get that number, i.e.,

`-log`

_{a}(b) = n

log_{a}(1/b) = n

or

`1/a`

^{n} = b

Negative logarithms are frequently used in analytical chemistry to determine the pH of aqueous solutions.

Also, remember that the **negative logarithm of a number and logarithm of a negative number is not the same thing**, i.e.,

`-log`

_{a}(b) ≠ log_{a}(-b)

**The logarithm of a negative real number is not defined**. For details on finding the logarithm of complex numbers, you can refer to this github post.

## An example of negative log calculation

To demonstrate how to find the negative log of any number using our online **negative log calculator**, let us calculate the value of `-log`

:_{2}^{}(8)

- Enter the number you want to calculate the negative logarithm of, i.e., 8 in the first row.
- Enter the base, i.e., 2 in the second row.
- If you want to calculate the natural log, use e as the base.
- The negative log value, i.e., -3, appears in the last row.

## FAQ

### Can log be negative?

**Yes**, the log of a number may be positive, negative, or even zero.

### Can you take the log of a negative number?

**No**, you can't take the log of a negative number. As discussed earlier, the log function `log`

is the inverse of the exponent function _{a}(b) = n `a`

, where the base ^{n} = b`a > 0`

. Since, the base *a* raised to any exponent *n* is positive, the number *b* must be positive. **The logarithm of a negative number b is undefined.**

### Can the base of a log be negative?

**No**. The base of a log function is also the base of an exponential function. If we raise a negative number (for example, -2) to any rational number that is not an integer (say, 1/2), we might end up with an imaginary number (`(√2)i`

). Since logarithms are defined for real numbers, **the base of a log function must be positive**.

However, it is possible to determine the log of an imaginary number or a negative number using Euler's identity.