Introduction to Logarithms

A mathematical approach to find the number of multiplying factors when any number is divided as some number of multiplying factors on the basis of another number, is called the logarithm.

The logarithm represents the inverse operation of the exponentiation. The inverse operation of the exponentiation is symbolically represented by the $\log$, which is an abbreviation form the word logarithm.

Example

According to Exponentiation, on the basis of number $2$, the number $8$ is expressed as $3$ multiplying factors. The relation between three of them is simplify expressed in exponential notation.

$8 = 2 \times 2 \times 2$ $\implies 8 = 2^3$

The same mathematical operation is written inversely.

$\log_{2} 8 = 3$

The total number of multiplying factors is $3$, when the number $8$ is expressed as multiplying factors on the basis of number $2$. Hence, it is read as the logarithm of $8$ to base $2$ is equal to $3$.

The relation of the Logarithm with Exponentiation can be expressed mathematically as follows.

$8 = 2^3 \Longleftrightarrow \log_{2} 8 = 3$

Examples

No need to remember the number of multiplying factors for every number that is expressed as number of multiplying factors on the basis of another number. It can be easily determined from the operation of exponentiation.

$(1) \,\,\,\,\,\,$ $\log_{3} 9$

Take $\log_{3} 9 = x$ and express it in exponential notation.

$\implies 3^x = 9$

$\implies 3^x = 3^2$

Therefore, $x = 2$ by the comparison.

$\therefore \,\,\,\,\,\,\,\, \log_{3} 9 = 2$

It states that the total number of multiplying factors is $2$ when the number $9$ is split as multiplying factors on the basis of number $3$.

$(2) \,\,\,\,\,\,$ $\log_{4} 64$

$\implies \log_{4} 64 = y$

$\implies 4^y = 64$

$\implies 4^y = 4^3$

$\implies y = 3$

$\therefore \,\,\,\,\,\,\,\, \log_{4} 64 = 3$

It expresses that the total number of multiplying factors is $3$ when the number $64$ is divided as multiplying factors on the basis of number $4$.

$(3) \,\,\,\,\,\,$ $\log_{5} 15625$

$\implies \log_{5} 15625 = z$

$\implies 5^z = 15625$

$\implies 5^z = 5^6$

$\implies z = 6$

$\therefore \,\,\,\,\,\,\,\, \log_{5} 15625 = 6$

It says that the total number of multiplying factors is $6$ when the number $15625$ is split as multiplying factors on the basis of number $5$.

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