Use this Log Calculator for any type of logarithmic calculation. Also, it gives fast and accurate answers. Even more, the Logarithm calculator is absolutely free to use.

### Logarithm Definition

The logarithm can be defined as the inverse function to the exponent. The concept of the logarithm is used in various fields. In mathematics and science, you find the most. Almost every branch of science implements a logarithm.

Let ‘b’ be raised to the power of x is equal to y. The mathematical equation for it is:

**b ^{x} = y**

Then, the logarithm of y with base b. It will be equal to x. The mathematical equation for it is

log_{b}(y) = x

##### Example 1

We know that 2^{3} = 8. Then the

**log _{2}(8) = 3**

Here, the base is 2. So, the log of 8 with base 2 is 3. Do it in the logarithm calculator.

##### Example 2

We know that 2^{5} = 32. Then the

**log _{2}(32) = 5**

Here, the base is 2. So, the log of 32 with base 2 is 5.

This is how the log calculator works. There are some logarithmic rules. We will also see them. The log tells the number of common occurrences. Occurrences of the factor in repeated products.

##### Example 3

100 = 10 x 10 = 10^{2}. Therefore, logarithm base 10 of 100 is 2. Consequently,

**log _{10}(100) = 2**

##### Example 4

10000 = 10 x 10 x 10 x 10 = 10^{4}. Therefore, logarithm base 10 of 10000 is 4. Consequently,

**log _{10}(10000) = 4**

### History of Logarithm

John Napier founded the concept of Logarithm. It was an efficient way to simply big numbers. Also, made the calculation much easier. It was accepted by all professions. As it simplified various calculations. Time elapsed was very less. Also, you can find out accurate results. Now, you have to spend a lesser amount of time. Use our log calculator to find the logarithm of any number.

### Applications of Logarithm / Log Calculator

As we have learned about what is logarithm, how the logarithm calculator works. So, now it’s time to look over some applications. Why logarithm is used? The following are the main reasons:

- Logarithms can be best used for representing very big and very small numbers. This simplification makes the calculation easier. Furthermore, scientific quantities like pH implement this concept.
- Physical effects like cooling down of hot bodies and damping of the oscillator are best represented using logarithmic decrement.

### Logarithm rules

**1. Logarithm product rule**

log_{b} (x ∙ y) = log_{b} (x) + log_{b} (y)

**2. Logarithm quotient rule**

log_{b} (x / y) = log_{b} (x) – log_{b} (y)

**3. Logarithm power rule**

log_{b} (x^{y}) = y ∙ log_{b} (x)

**4. Logarithm base interchange rule**

log_{b} (c) = 1 / log_{c} (b)

**5. Logarithm base change rule**

log_{b} (x) = log_{c} (x) / log_{c} (b)

**6. Logarithmic derivative**

f (y) = log_{b} (y) ⇒ f ‘ (y) = 1 / (y ln(b))

**7. Logarithmic Integral**

∫ log_{b}(y) dx = y ∙ ( log_{b}(y) – 1 / ln(b) ) + C

**8. The logarithm of negative numbers**

The logarithm of a negative number is undefined. The graph of the log always goes in a positive direction.

**9. Logarithm of zero**

log_{b} (0) = undefined

**10. Logarithm of 1**

The log_{b} (1) = 0

**11. Logarithm of base**

The log of the base will be log_{b} (b) = 1

**12. Logarithm of Infinity**

lim log_{b} (x) = ∞ (Infinite), when x → ∞ (Infinite)

### How to use the Log Calculator?

A logarithm calculator is an online tool. You can use it just by opening its website. In any web browser. In any internet-accessible device. Anytime and anywhere. Even more, you can get rapid and accurate results. Follow these steps one by one. It will guide you through finding the log of a number.

At first, get a net connection. Then open your web browser.

- On your web browser, open the Log Calculator. Let it load completely.
- Select the base (2, e or 10). It is in the manner of a drop-down list.
- On the log calculator web page. You can see an option to enter the number of which you want to find out the logarithm. Enter the number there.
- Press the ‘Calculate’ button to find the log. As a result, the logarithmic result will be displayed below. There is also a copy option. Also, you can see the graph of your log function. Lastly, press ‘Reset’ for new log calculation.