Log Calculator

What is Log (Logarithm)?

Log is the inverse of the mathematical operation of exponentiation and is also known as the logarithm. Exponentiation is raising a base number to a power, while logarithm determines the exponent needed to reach a given number.

In simpler words, the log or logarithm tells what power we must raise on a specific base to get a given number. If “b” raised to power “y” that gives resultant “x”, then the logarithm of “x” with base “b” is equal to “y”. It is mathematically represented as:

If b^y = x ⟺ log_b(x) = y

Where:

• “b” is the base of logarithm (which must be greater than 1)
• “x” is the required number (must be positive)
• “y” is the exponent of the log.

Logarithm Types

There are three common types of logarithms due to their different bases, which named as:

• Common Logarithm
• Natural Logarithm
• Binary Logarithm

• Common Logarithm

The log of any number with base 10 is known as the common logarithm or log base 10 and represented as log₁₀ or log. Common logarithm simply defines how many times multiply the number 10 to get the required result.

I.e., log₁₀(100)=2, it means we multiply the number “10” twice to  get “100”.

• Natural Logarithm

The logarithm of any number with base “e” is known as the natural logarithm or base e logarithm. It shows by ln or log_e, here “e” is euler’s constant, whose value is equal to 2.71828. 

• Binary Logarithm

The logarithm’s with base 2 are known as the binary logarithm or log base 2 and mathematically represented as “log₂^(x)”. It tells how many times 2 is multiplied to get a required number. 

Log Rules

Here, we explain some important log rules and properties that help to perform logarithmic operations easily, such as by writing multiplication & division of numbers in addition or subtraction of logarithms.

• Product Rule: This rule is used to solve the logarithm of a product of two numbers. It converts the multiplication of two logarithmic values into the sum of their individual logarithms. 
                              log_b (A * B)= log_b(A) + log_b(B)
• Quotient Rule: It is used to solve the logarithm of 2 division values. It converts the log of two division values into the difference of each logarithm.
                              log_b(A/B) = log_b(A) - log_b(B)
• Power Rule: In this exponential rule, the logarithm of “A” with rational exponent “m” is equal to the exponent (m) times of log of a given number.
                              log_b(A^m)= m×log_b(A)
• Change of Base Rule: This rule alters the logarithm’s base to any other dummy base.
                              log_b(A) = log_c(A)/log_c(b) (for any base c)
• Base Switch Rule: This rule switches the base with an exponent of the given logarithm.
                              log_b(a) = 1/ log_a(b)
• Log of 1: The log of 1 for any base is always 0, such as: log_b(1)= 0.
• Log of same Base: If the base and exponent of the logarithm are same then its log value is always 1. i.e., log_b(b)=1.

How to Calculate Log?

The best method is to use our logarithm calculator. But if you find it manually, then calculate log by below steps:

• Identify the base (b) and the exponent “x” from the given logarithmic expression.
• If the base of given expression is “10” then use the common logarithm table. If base is “e” then use the natural logarithm table. However, for any base value use our above log base calculator.
• After selecting a table according to base, then see the value of the exponent from the selected log table.
• Finally, writing the required log value by using logarithmic rules.

Alternatively, you can also use the logarithmic rule for some specific problem (for base 10) to calculate log. See the below example to understand how to calculate log without the use of any logarithmic calculator or log table.

Example 1: Find the value of “log₁₀⁽¹⁰⁰⁰⁾”.

Solution:
To calculate logarithm of that type of number, we find the factors of the exponent and apply the log rules for their final value.

Step 1: First, we find factors of “1000”.

1000 = 10×10×10 = 10³

Step 2: Now, put value in log expression.

log₁₀⁽¹⁰⁰⁰⁾= log₁₀^{(10^3)}

By using the power rule of logarithms, we get.

log₁₀⁽¹⁰⁰⁰⁾= 3 log₁₀⁽¹⁰⁾

Now, use the log rule of same base (i.e., log₁₀⁽¹⁰⁾= 1) and get.

           log₁₀⁽¹⁰⁰⁰⁾= 3 (1) = 3

To verify the above result, use our log calculator.

Example 2: Calculate log of “24” for base 10 by using logarithm table.

Solution:

The log conversion of any number with a logarithm table having two parts (characteristics & mantissa). Then we find both values separately. 

Step 1: Find the characteristic.

A characteristic is total number of moving digits in the placement of decimal point after the first non-zero digit. It is a positive number if decimal moves from right to left & “–ve” moves from left to right.

Thus, characteristic of “24” is 1.

Step 2: Now, find the mantissa by using a logarithm table.

For this, we see row of “24” in column “0” of the common log table and get a value of 0.3802.

So, mantissa = 0.3802

Step 3: Now, add characteristic + mantissa to find final value.

log₁₀⁽²⁴⁾= 1+0.3802 = 1.3802

Thus, log of 24 with base 10 is “1.3802”. Moreover, if you want to find the logarithm without using any log table, then use our above logarithms calculator.