Log Base 2

In Mathematics, before the calculus era, many mathematicians were using the logarithms to change multiplication and division problems into the addition and subtraction problems. In the Logarithms, the power is raised to some numbers, which is usually the base number. Then it can give some other number. It is the inverse function of the exponential function. It’s one format is very useful in the computer field also, and that is the Log Base 2 or Log 2 or Binary base logarithm. This article will explain log base 2 in a very simple way with examples. Let us learn it!

Log Base 2

 

Source: en.wikipedia.org

Logarithm Function:

In Maths, the logarithm is computed as the inverse function of the exponentiation. In easy terms, the logarithm is defined as the power to which a number must be raised in order to get some other value. The general definition of it is as follows:

\(log_{a}(y)\) = x

where the base is a and value is x. Its equivalent expression in exponent form is:

\(a^{x}\) = y

Common Logarithmic Function:

The logarithmic function with base 10 is very popular and termed as the common logarithmic function. It is denoted by \(log_{10}\) or simply log. Thus,

f(x) = \(log_{10} x\)

Natural Logarithmic Function:

The logarithmic function to the base e is known as the natural logarithmic function. It is denoted as \(log_{e}\) or in short ln. Thus.

f(x) = \(log_{e} x\)

Here, e is the number which is an irrational and transcendental number, with an approximate value equal to 2.718281828459…

log base 2:

Log base 2 or log 2 is also termed as a binary logarithm. It is the inverse function of the power of 2 functions. The general logarithm says that for every real number N, we have exponential form as

N = \(a^{x}\)

Here, a is the positive real number and called as the base here. And, x is the exponent. Then it is written as:

\(log_{a} N\) = x

Log base 2 is the power to which the number 2 is raised to get the value of N. For any real number x, log base 2 functions will be:

x = \(log_{2} N\)

Which is same as:

\(2^{x}\) = N

It must be noted that the logarithm of base 0 does not exist. Also, logarithms of negative values are not defined in the real number system. Log base 2 table is very useful in the calculations.

The formula for Change of Base:

The logarithm is in the form of a log at base 10 or e or any other bases. We may change the base by using the following formula.

\(\log _{2}\)x=\(\frac{\log_{10}x}{\log_{10}2}\)

It is based on the general formula for change of base as:

\(\log _{a}x\)=\(\frac{\log_{b}x}{\log_{b}a}\)

Thus, to get the value of log base 2, we need to convert it into common logarithmic functions at base 10 by using the change of base formula.

Properties of Log Base 2:

Some of the log base 2 properties are as follows:

• Product Rule: \(log_{2} (x \times y)\) = \(log_{2} x + log_{2}\) y
• Quotient Rule: \(log_{2} (\frac{x}{y})\) = \(log_{2} x – log_{2}\) y
• Power Rule: Raise the exponential expression to the power and multiply the exponents. Thus,

\(log_{2} x^{n}\) = \(n log_{2} x\)

• Zero Exponent Rule: \(log_{a} 1\) = 0.
• Change of Base Rule: \(log_{b} x\) = \(\frac {log_{2} x} { log_{2} b}\)

This formula is useful to change log 2 to log 10 also.

Solved Examples for You

Q.1: Find the values of following logarithm terms.

1. log 1 base 2
2. log 7 base 2
3. 3 log base 2
4. log 2 base 2

Solution: (i) log 1 base 2 = \(log_{2}\) 1 = 0 . It is true for any base.

(ii) Log7 in base 2

i.e. \(\frac { log7}{log2}\). It will be 2.80735492205760.

(iii) 3 log base 2

i.e. \(log_{2} 3\) = \(\frac { log3}{log2}\). It will be 1.584963

(iv) log 2 base 2

i.e. \(log_{2} 2\) = 1