Logarithm Formula

Logarithms are widely used in computations in mathematics as well as in science. It helps to solve complex problems involving exponents of variables, easily. Many derivations of physics are possible only due to Logarithm Formula. A logarithm is the inverse computation process of exponential. Logarithms are widely used in the field of physics, chemistry, biology, computer, etc. We can even find logarithmic calculators which have made our calculations much faster and easier. These find many applications in surveying and celestial navigation purposes. To understand the concept of the logarithm, let us have this useful article. We will see various logarithm formula in mathematics with examples and their applications.

Logarithm Formula

What is Logarithm?

As we know that \(3^4\)=81

Now suppose if we are asked the same question but differently, like “what will be the exponent of 3 to get the result 81?”

Then obviously answer will be 4. But how? The answer to this question only is the basic definition of logarithms.

Now, we will write the above equation in the form of a logarithm, as

\(\log_{3}\) 81 = ?

Here, 3 is the base whose exponent we have to find. So we wish to find the value which when rose as the power to 3 will be equal to 81. Since this will be 4, so we will say that

\(\log_{3} 81\) = 4

This above equation will be read as “log base 3 of 81 is 4”.

Thus ,general definition and rule of logarithm is:

\(\log_{a} x = b ⇒ x = a^b\).

Hence, the exponent or power to which a base must be raised to yield a given number is nothing but the logarithm.

Also,

Since 10² = 100,

Then we get

\(2 = \log_{10} 100\)

Logarithms with the base 10 are called common or Briggsian logarithms and it is written simply log n. It is invented in the 17th century to speed up calculations. The natural Logarithm is with base e where e ≅ 2.71828, and it is written as ln n.

Logarithm Formula:

Two most trivial identities of logarithms are:

(1) \(\log_{b} 1 = 0\)

This is because  \(b^0\) = 1;

(2) if b>0 then

\(\log_{b} b = 1\)

This is because \(b^1=b\)

Some other very important formula are:

Suppose a, b , m, n are variables with positive integers and p as a real number. Then we have,

(1) \(\log_{b} m*n  = \log_{b} m +  \log_{b} n\)

(2) \(\log_{b} \frac{m}{n}  = \log_{b} m +  \log_{b} n\)

(3) \(\log_{b} n^p = p \log_{b} n\)

(4) \(\log_{b} n = \log_{a} n * \log_{b} a\)

Solved Examples

Q.1: Solve \(\log_  {64}\) = ?

Solution- since \(2^ 6 = 2 * 2 * 2 * 2 * 2 *  2\)

i.e. \(2^6\) = 64,

Thus 6 is the exponent value.

So, \(\log_2 {64}\) = 6.

Q.2:  By using property of logarithms, solve for the value of x

\(\log_3 x=\log_3 4+ \log_3 7\)

Solution- Here for RHS we will use the addition rule of logarithms.

\(\log_3 x =\log_34+ \log_3 7\)

Then \(\log_3 x = \ log_3(4 * 7 )\

Thus value of x is = 28.