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What is a Log (Logarithm)?

The logarithm is the inverse of the exponent function. By inverse, it means a function that does the opposite of the exponent function. For making it easier to understand for you, consider some examples like subtraction is the inverse of addition and division is the inverse of multiplication. Where exponent means to multiply a number “x” times like 2x is the as taking “x” number of 2s and multiplying them together.

Logarithm tells you ways repeatedly you have got to multiply the base together to get the number. For example, log2 (x) is counting how many 2s would need to be multiplied to make x.

Examples of Log Function Using Actual Numbers

24 = 16
Log2 (16) = 4
These both deal with this series of multiplication
2x2x2x2

The exponent takes the count and multiples to make a number. On the other hand, logarithms take the final number and determine the count of multiplication. The log function is all about repetition of numbers.

Common Factors in Exponents and Logarithms

In exponents and logarithms both, we deal with the “base”. The “base” of the exponent will be the same as the base of the log. Fortunately, the base is always the lowest number you write in both cases. By lowest, it means vertically lowest not lowest as in the least number.

Such as 4x and log4 (x) are both base 4
These bases are directly connected
If 4x = y then log4 (y) = x

What is the difference between a Natural Logarithm and a Base Ten Logarithm?

Natural logarithm (abbreviated as ln), is a logarithm to base e. That is, applying the ln function on a number x gives the answer to what power do I need to raise e to, to get x?

If you are unfamiliar with “e”, it's basically a constant roughly equal to 2.718. But what makes the constant special enough to own a singular name for its logarithm? What makes special, is that the exponential function ex is its own derivative. In fact, that's the definition of “e”. The logarithm base “e” is the inverse of this function, and because of this unique property of ex, is rather important in calculus.

In perspective on the way types' work, you can truly set up a logical association between the regular logarithm, and the logarithm to base 10: Ln(x) ≈2.302∗log10(x)

This works because of ln (10) ≈2.302

What are the Basic Properties and Rules for Logarithms?

Since taking a logarithm is the opposite of exponent function (more precisely, the logarithmic function logb x is the inverse function of the exponential function bx), we can drive the basic rules for the logarithms from basic rules for exponents. For straightforwardness, we'll compose the standards as far as the characteristic logarithm ln(x). The rules apply for any logarithm logb x, except that you have to replace any occurrence of “e” with new base “b.” As per the formula and definition of the natural log which we discussed earlier, we determine that a relationship between the natural log and the exponential function is

S eln c = c …… 1
By combining the values of “c”, we get
Ln (ek) = k …….. 2

These equations simply that ex and ln x are inverse functions. So we drive the following rules or properties for the logarithms based on equation 1 and 2.

1. Product Rule

ln (xy) = ln(x)+ln(y)

2. Quotient Rule

ln (x/y) = ln(x)−ln(y)

3. Log of Power

ln (xy) = yln(x)

4. Log of e

ln (e) = 1

5. Log of One

ln (1) = 0

6. Log of Reciprocal

ln (1/x) = − ln(x)

A Brief Overview of Logarithms’ History

By simplifying difficult calculations, logarithms contributed to the advance of science and especially of astronomy. They were critical advances in surveying, celestial navigation and other domains. Pierre Simon Laplace called logarithm. A key tool that enabled the practical use of logarithms before calculators and computers were the tables of logarithms. The first such table was compiled by Henry Briggs in 1617, after Napier’s Invention.

  • Subsequently, tables with increasing scope and precision written
  • These tables listed values of logb (X) and bx for any number “x” in a certain range, at a certain precision, for a certain base “b” which is usually 10
  • For Example, Briggs’ first table contained the common logarithms of all integers in the range
    1- 1000, with a precision of 8 digits
  • As the function f(x) = bx is the inverse function of logb (X), it has been called the antilogarithm

Applications of Logarithms

You can find as many applications of logarithms as you want both inside and outside mathematics. But in most cases, logarithms are used in

  • Logarithmic Scale
  • Fractals
  • 2 Digit Expense Calculation
  • Finding Order of Magnitude
  • Calculating Interest Rates
  • Logarithmic Graphs

What is a Log Calculator?

A log calculator can be found with many names online such as logarithm calculator, log base 2 calculators, log base calculator or natural log calculator. Sometimes, it is even called as log solver but the purpose of all these calculators is the same. A logarithm calculator cuts of your hassle into the half when it comes to calculating and finding logs of different numbers. There are several log calculators which you can find online to solve logarithm problems but our tool outshines every other natural log calculator in terms of ease of usability and free of cost usage.

How to use a Log Calculator?

Our log calculator is as simple to use as one can find. You don’t need to be a master to get the best out of it. You only need to perform two steps to get log value of your desired digits

  • First Enter the Value
  • Select between the base of 2, e and 10
  • Click on the “Calculate” Button

You will get the results right after clicking on the calculate button. This makes our log calculator also one of the fastest log calculators you can find online.


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