Number
Base
Negative logarithm

# Negative Log Calculator

By Purnima Singh, PhD
Last updated: Apr 01, 2021
Table of contents:

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; `23 = 8 `, i.e., you have to multiply three 2's together to get 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 `an = b` where 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:

`loga(b) = n `

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

We know that `23 = 8 `, i.e., `log2(8) = 3`. Hence, we can say that the logarithm of 8 to base 2 is 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.,

```-loga(b) = n loga(1/b) = n ```

or

`1/an = 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.,

```-loga(b) ≠ loga(-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 `-log2(8)` :

1. Enter the number you want to calculate the negative logarithm of, i.e., 8 in the first row.
2. Enter the base, i.e., 2 in the second row.
3. If you want to calculate the natural log, use e as the base.
4. 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 `loga(b) = n ` is the inverse of the exponent function `an = b`, where the base `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.

Purnima Singh, PhD