Log / Logarithm Calculator

Use this log calculator or logarithm calculator to calculate a logarithm of a specific number with an arbitrary base. Whether you’re looking for a natural logarithm or a common logarithm, this logarithm calculator will do the computation for you. In this article, we’ll learn how to use the calculator along with other relevant information you need to understand logarithms better.

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How to use the log calculator? 

Using this log base calculator is a simple and easy task. Instead of having to compute for the logarithms, you enter the required values. Just like other online calculators, this one automatically generates the value you need in an instant. Here are the steps to follow when using this natural log calculator:

  • First, enter the value of the logarithm you’re looking for on the space “Logarithm of.”
  • Next, enter the value of the “Logarithm Base.”
  • Upon entering both required values, the log calculator will generate the Logarithm Value you need automatically.


What is a logarithm? 

A logarithm is the inverse of an exponent. Therefore, a logarithmic function refers to the inverse of an exponent function. If you raise a logarithm to power y and it gives you x, this means that the log of x with a base y is equal to x. This may seem confusing when described in words so let’s take a look at the formula:

  ay = x is equivalent to log_a(x) = y

When calculating the natural log of any given number, you have to select a base that’s equal to “e” or 2.718. One of the most common log base used in mathematical computations is 10. A log with this base is a “natural log.” The symbol ln(x) represents a natural log.

There are times when you need to calculate a log which has an arbitrary base. Most of the time though, you only have access to a natural log calculator or a logarithm calculator with a log base of 10. In such a case, you need to keep these rules in mind:

  • log_a(x) = ln(x) / ln(a)
  • log_a(x) = lg(x) / lg(a)

Apart from these, there are a few basic logarithm rules for log operations. For you to understand all concepts related to logarithms, you need to know these rules too. For your reference:

  • log_a(x*y) = log_a(x) + log_a(y)
  • log_a(x/y) = log_a(x) – log_a(y)
  • log_a(x^y) = y*log_a(x)


What is natural logarithm? 

What is natural logarithm? 

In exponential logarithms and functions, they can have a base of any number. But there are two specific bases which are so frequently used that mathematicians have come up with unique names for these logarithms. These are the natural and common logarithms.

A natural logarithm of any number is the number’s logarithm to the base of “e,” which is a mathematical constant. Here, “e” refers to a transcendental and irrational number that’s equal to 2.718281828459. When writing the natural log of x, you write it as ln x or loge x. In some cases, when the number has an implicit base “e,” write it as log x. You may also add parentheses to make things clearer giving you ln(x), loge(x) or log(x).

Another definition of a natural log is “any positive real number “a” as the area under the curve y = 1/x from 1 to a (the area being taken as negative when a < 1).” This is a simpler definition which matches other formulas which involve natural logs.

You can also extend the definition of a natural log to provide log values for all the non-zero complex numbers and negative number. However, this leads to a function with multiple values known as a complex logarithm.

When you consider a natural log function as a real-valued function of a real variable, it becomes the inverse function of an exponential function. Most types of calculators come with buttons for Log and Ln which symbolizes log base 10. When you use a calculator, you can compute the logs in base 10 or base “e” with a single click.


What is a common logarithm? 

Common logs are also known as “Briggsian logarithms” after the British mathematician from the 17th-century known as Henry Briggs. Back in 1616 to 1617, Briggs worked with John Napier who invented the natural logarithms. He suggested that Napier make a change to his logarithms.

During these meetings, they both agreed upon the modification that Briggs proposed. After coming back from one of his visits, he published the first part of his logarithms.

In mathematics, a common log refers to the log with a base 10. It also has other names which are the “decimal logarithm” and the “decadic logarithm.” You can indicate the common logarithm with the symbol log10(x) or Log(x). But the latter is an ambiguous notation because it can also stand for a complex log.

If you try to find this function on a calculator, it’s the “log” function. But most mathematicians mean the natural log whenever they write “log.” To lessen this ambiguity, instead of writing log10(x), write it as lg(x). Also, instead of writing loge(x), write ln(x).

Common logs are any logarithms with a base 10. Remember that the number system is base 10. The system has 10 digits and you determine place value by groups of 10. This makes it easier for you to remember what common logs are.


How to expand a logarithm? 

When you take the quotient rule, the power rule, and the product rule together, they’re called “properties of logarithms.” Sometimes, you need to apply more than one of these rules for you to expand a logarithm. For instance: 

logb(6x/y)        =logb(6x)−logby



Use the power rule for expanding logarithms which involve fractional and negative exponents. There also exists an alternate proof of the quotient rule which uses the fact that reciprocals are negative powers:


logb(AC)         =logb(AC−1)




Then you can use the product rule when expressing a difference or sum of logs as the log of a product. Although it may seem confusing at first, practising allows you to look at any logarithmic expression, mentally expand it, then write down the final answer. Keep in mind though, that you may only do this with roots, quotients, powers, and products. However, you can’t do this with subtraction or addition inside the logarithm’s argument.