Value of log 2 base 10 is 0.301 and value of log 2 base e is 0.693147. Log function (logarithmic function) is a mathematical function that is used to reduce or limit the complexity of equations. A logarithm (or log) to the base b of a number x is a mathematical function that tells us the number n to which the “base” number b must be raised to generate the number x. Mathematically, a logarithm can be written as: \(log_{b}x\) = n then \(b^n = x\).

Logarithms can be classified into 2 types:

1. Common Logarithmic Function: Base 10 logarithm is called a Common logarithm. The symbol for it is log.
2. Natural Logarithmic Function: Base e logarithm is called a Natural logarithm. The symbol for it is ln.

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What is Value of Log 2?

Value of log 2 to base 10 is 0.301 and the value of log 2 to the base e is 0.693147. Logarithms are the other way of writing exponents. A logarithm of a number with a base is equal to another number. Hence, we can conclude that \(log_{b}x\) = n or \(b^n = x\) where b is the base of the logarithmic function. This can be read as “Logarithm of x to the base b is equal to n”. Thus, log 2 is written as \(log_{10}2\) = 0.301. We can use the value of log 2 to find the logarithm of any number. The simplest way to find the value of the given logarithmic function is by using the log table. In case you don’t have a log table, here is a method that can be used to calculate the logarithm of a number. If the number is too huge to be calculated using a calculator, this method can also be used. First and foremost, keep in mind the following basic log values (it’s not a bad idea to keep these in mind even if you have a log table).

\(log_2\) = 0.301

\(log_3\) = 0.477

\(log_5\) = 0.699

\(log_7\) = 0.845

These four numbers can be used to calculate the log of any number (of any size), and the formula is as follows:

\(log(a+b) = log(a) + {b\over{(2.42×a)}}\)

This is the main method to find any log value.

The common logarithm has base 10 and is represented on the calculator as log(x). The natural logarithm has base e, a famous irrational number, and is represented on the calculator by ln(x). The natural and common logarithm can be found throughout Algebra and Calculus. Let’s find the value of common and natural of 2.

Value of \(log_2\) in Common Log

The log function of 2 to the base 10 is represented as “\(log_{10}2\)”.

According to the definition of the logarithmic function,

Base, b = 10 and \(10^n\) = b

With the use of a logarithm table, the value of log 2 to the base 10 is given by 0.3010.

\(log10_2\) = 0.3010.

We can also find the value of log 2 by approximation.

We know that,

log 10 = 1

log 5 + log 2 = 1

0.5 log(25) + log 2 = 1

0.5 (log 2.5 + log 10) + log 2 = 1

Say log 2.5 is nearly equal to log 2.

Then, 1.5 log 2 = 0.5

log 2 = 0.3.

Value of \(log_2\) Natural Log

The natural log function of 2 is represented as “\(log_e2\)”. The log function of 2 to the base e is another name for it. The natural log of 2 is represented by the symbol ln (2). The \(log_e2\) coefficient is 0.693147.

\(log_e2\) = ln (2) = 0.693147

The natural logarithm of 2 is a transcendental variable that is frequently encountered in decay problems, particularly when converting half-lives to decay constants. ln2 = 0.69314718055994530941 is the numerical value of ln2.

How to Find the Value of Log 2

Let’s learn how to find the value of log 2 using two common types of logarithms: the common logarithm and the natural logarithm.

1. Value of log 2 in Base 10 

The common logarithm is the log with base 10. It's written as:

log₁₀(2)

This means: “What power should we raise 10 to, to get 2?”

Using a log table or calculator, we find that:

log₁₀(2) = 0.3010

So, 10 raised to the power of 0.3010 is equal to 2.

2. Value of log 2 in Base e 

The natural logarithm uses base e (where e ≈ 2.718). It’s written as:

ln(2) or logₑ(2)

This means: “What power should we raise e to, to get 2?”

Using a calculator, the value is:

ln(2) = 0.693147

So, e raised to the power of 0.693147 is equal to 2.

Log Base 2 – Basic Rules and Properties

Logarithms with base 2 follow the same rules as other logarithms. Here are the key properties of log base 2 explained simply:

1. Log of 1 is Always 0

If you take the log of 1 with base 2:

log₂(1) = 0

This is because 2⁰ = 1

2. Log of a Number and Its Own Base is 1

If the number and the base are the same:

log₂(2) = 1

Because 2¹ = 2

3. Adding Two Logs Means Multiplying the Numbers

When you add two logs with base 2:

log₂(a) + log₂(b) = log₂(ab)

So, you can combine the logs by multiplying the numbers inside.

4. Subtracting Two Logs Means Dividing the Numbers

When you subtract one log from another:

log₂(a) − log₂(b) = log₂(a / b)

You can combine them by dividing the numbers inside.

5. Changing the Base to Log Base 10 (or Any Other Base)

You can change log base 2 into another base using this formula:

log₂(N) = log(N) / log(2)

For example, if you're using a calculator that only has log base 10, you can still find log base 2 this way.

6. Log of a Power Means You Can Bring the Power to the Front

If a number has an exponent, you can move the exponent in front of the log:

log₂(nᵏ) = k × log₂(n)

This is helpful when dealing with powers and makes solving easier.

 Summary Table:

Value of Log from 1 to 10

Value of Log from 1 to 10 can come in handy for finding out the values of the logarithm of bigger numbers. The values from log 1 to 10 to base 10 are given below:

Value of Ln from 1 to 10

Value of Ln from 1 to 10 can come in handy for finding out values of the natural logarithm of bigger numbers, just like the table of the common logarithm. The value of ln 1 to 10 in terms of the natural logarithm \(log_ex\) is listed here.

Logarithm Function

A logarithm is the opposite (or inverse) of an exponent.

If you know that:
aᵡ = b,
then the logarithm tells you:
logₐ(b) = x

This is how we write the logarithmic function:

F(x) = logₐ(x)
Here, a is the base of the logarithm, and x is the number.

Rules of Logarithms

To solve problems easily, you need to remember a few basic log rules:

 Product Rule

When multiplying inside the log, add the logs:

log_b(M × N) = log_b(M) + log_b(N)

 Quotient Rule

When dividing inside the log, subtract the logs:

log_b(M ÷ N) = log_b(M) − log_b(N)

 Power Rule

If a number inside the log is raised to a power, bring the power in front:

log_b(Mᵖ) = p × log_b(M)

 Zero Exponent Rule

This is a reminder that:

logₐ(a) = 1
Because a¹ = a

 Change of Base Rule

If you want to change the base of a logarithm:

log_b(x) = log₁₀(x) / log₁₀(b)
Or
log_b(x) = ln(x) / ln(b)

Use this rule when your calculator only supports log or ln functions.

Solved Examples on Value of Log 2

Now let’s see some solved examples based on Value of Log 2.

Example 1: Find out the value of \(log_5512\) if \(log_{10}3 = 0.4771 \) and \(log_{10}2 = 0.3010\).

Solution: \(log_5512\)

\(= log_52^9\)

\(= {9log2\over{log5}}\)

\(= {9log2\over{log{10\over2}}}\)

\(= {9\times0.3010\over{log10 – log2}}\)

\(= {9\times0.3010\over{1 – 0.3010}}\)

\(= {9\times0.3010\over{0.6990}}\)

= 3.876

Thus, \(log_5512\) = 3.876

Example 2: Simplify \(log_{10}18\) and \(log_{10}54\) if \(log_{10}3 = 0.4771\) and \(log_{10}2 = 0.3010\).

Solution: \(log_{10}18\)

\(log_{10}18 = log_{10}(9\times2)\)

\(= log_{10}9 + log_{10}2\)

\(= log_{10}3^2 + log_{10}2\)

\(= 2log_{10}3 + log_{10}2\)

\(= 0.3010 + 2(0.4771)\)

\(= 0.3010 + 0.9542\)

= 1.2552

Thus, \(log_{10}18\) = 1.2552

\(log_{10}54\)

\(log_{10}54 = log_{10}(2\times27)\)

\(= log_{10}2 + log_{10}27\)

\(= log_{10}2 + log_{10}3^3\)

\(= log_{10}2 + 3log_{10}3\)

\(= 0.3010 + 3(0.4771)\)

\(= 0.3010 + 1.4313\)

= 1.7323

Thus, \(log_{10}54\) = = 1.7323

Example 3: The value of \(log_a1 + log_22^2 + log_33^3log_a1 + log_22^2 + log_33^3\) (where a is a positive number and a ≠ 1) is

Solution: \(log_a1 + log_22^2 + log_33^3log_a1 + log_22^2 + log_33^3\)

\(= log_a1 + 2log_22 + (3log_33)log_a1 + 2log_22 + 3log_33\)

\(= log_a1 + 4log_22 + (3log_33)(1 + log_a1)\)

\(= log_a1 + [4{(log_{10}2\over{(log_{10}2}}] + [3{(log_{10}3\over{(log_{10}3}}](1 + log_a1)\)

\(= log_a1 + 4 + 3(1 + log_a1)\)

\(= log_a1 + 4 + 3 + 3log_a1\)

\(= 4log_a1 + 7\)

\(= 4({log_{10}1\over{log_a}}) + 7\)

= 7

Thus, \(log_a1 + log_22^2 + log_33^3log_a1 + log_22^2 + log_33^3\) = 7

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