Online Logarithm Calculator

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Log Calculator

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Learn About log calculator

You may use this log calculator to get the logarithm of a (positive real) integer with a specified base (positive, not equal to 1). This tool will resolve your issue whether you're seeking for a natural logarithm, a log base 2 log, or a log base 10 log.

Continue reading to learn more about the logarithm formula and the guidelines you must adhere to. Additionally, you could discover some intriguing knowledge, such as why and how logarithms are used in our daily lives.

Consider checking out our cube root calculator if you're looking for another useful math calculator that allows you to calculate not only the cube root but also roots of any degree.

What is a logarithm?

The opposite of an exponential function is a logarithmic function. In essence, the logarithm of x with base an is equivalent to y if a raised to power yields x. aʸ = x is the mathematical equivalent of logₐ(x) = y.

In other words, the logarithm of x, or logₐ(x), indicates what power we need to increase a to in order to generate the number x (or, if x is more than 1, how many times a should be multiplied by itself). From this perspective, we may also describe the logarithm as follows:

aloga(x) = x

Hopefully, you now know what a logarithm is, and you may learn about its two most common versions in the section that follows.

the common and natural logarithms

There are many possible bases for logarithms, but two in particular are used so frequently that mathematicians have given them their own names: the natural logarithm and the common logarithm.

Natural logarithm

If you wish to calculate the natural logarithm of a number, you must use a base that is roughly equivalent to 2.718281. The sign for this number is often the letter e, which honours Leonard Euler, who established its meaning in 1731. As a result, the logarithm can be shown as logex, however it is often represented by the sign ln (x). In particular in banking and economics, you could also find log(x), which also refers to the same function.

y = logₑx = ln(x) ,
where x = eʸ = exp(y)

Compound interest is a useful example of how to grasp how the natural logarithm works in real life. This interest is computed on the principal as well as the total interest. The following is the formula for yearly compound interest:
A = P(1 + r/m)ᵐᵗ

  • A is the investment
  • P initial value;
  • r is the interest rate
  • m times the interest
  • t is years.

Assuming you make a one-year deposit at a bank where interest is regularly compounded, m will be a huge amount. If you contrast frequencies with yearly (m=1), monthly (m=12), daily (m=365), or hourly (m=8,760) frequencies, it is simple to notice how rapidly the value of m is rising. The m would have increased significantly if your money were recalculated every minute or second.

Common logarithm

The common logarithm with a base of 10 is known as log₁₀x and is the other widely used type of logarithm (x). The Briggsian logarithm, which was created by the English mathematician Henry Briggs, is another name for it. Other names include the decimal logarithm, decadic logarithm, standard logarithm, and the logarithm.

It is the most popular type of logarithm, as its name implies. For instance, our decibel calculator makes use of it. In the past, common logarithms were also frequently included in logarithm tables that were intended to make computation easier.

How can I calculate a logarithm using any base?

The following guidelines must be followed if you only have access to a natural logarithm or a log 10 base calculator and you need to compute a logarithm with any base:

  • logₐ(x) = ln(x) / ln(a)
  • logₐ(x) = lg(x) / lg(a)
An example using log base 2

Assume for the moment that you want to employ this device as a log base 2 calculators. Any number's logarithm may be calculated by simply following these easy steps:

  1. Choose the number whose logarithm you wish to determine. Suppose it is 100.
  2. Choose your base; in this example, it is 2.
  3. Find the base-10 logarithm of the number lg(100) = 2.
  4. Find the base-10 logarithm of the number 2 lg(2) = 0.30103.
  5. Divide these numbers by one another to get the result 6.644: lg(100)/lg(2) = 2 / 0.30103
  6. You may simply enter the number and base directly into the log calculator by skipping steps 3-5.

Logarithms are used in arithmetic calculations.

Prior to the public's widespread availability to pocket calculators around the end of the 1970s, doing computations, particularly those involving fractions, required a significant amount of manual labor. The use of logarithms provided a useful purpose by relieving this tiresome effort. We must be familiar with the logarithm's fundamental features in order to take use of its technical benefit. These requirements should already be met, but just in case, the table below lists them.

Slide regulations

Further research was sparked by the new mathematical tool's rising appeal. The calculating line of the logarithm, a practical tool for division and multiplication, was invented by Edmund Gunter in 1620.

William Oughtred invented the first iteration of the gadget, which needed two compasses for measurement. This occurred in 1622. He created the traditional slide rule, which consists of two rulers that slide next to one another.

He developed a fresh method for further simplifying the computation of the law of logarithm in this way. Slide rules have become a common tool for calculation in fields requiring math. Up until the start of the Digital Revolution, scientists, engineers, architects, and even astronauts relied on their calculative aid. Slide rules were used by Albert Einstein, and the Apollo mission crews also brought them into space.

Real-world applications of logarithms

Current computers and scientific calculators have taken the role of antiquated methods in the modern day. Nevertheless, comprehending the ideas behind logarithms might improve your mathematical abilities. In several disciplines, logarithms are still useful in many practical ways

Logarithms connect arithmetic progression to geometric progression, which raises the possibility that some events in the actual world might follow a logarithmic pattern. There are many instances of the magical logarithm in both nature and everyday life, therefore this is true.


What is log1?

No matter the basis of the logarithm, the logarithm of one is always zero: For any a, loga 1 equals 0.

Can a negative log exist?

Depending on what you mean when you say "negative log," you may or may not be able to have one:

  • We can calculate a log's opposite: -logₐ(x) = logₐ(1/x)).
  • The log of a negative number, however, cannot be calculated.

Is log the same as ln?

No, log and ln are often not interchangeable. The symbols used in conventional mathematics are:

Sometimes lg stands for the logarithm with base 2, especially in publications discussing the binary system. ln stands for the natural log (thus with base e); log for the logarithm with base 10;