Question

Express in logarithmic form: ${5^3} = 125$

Answer 1

Hint : The logarithm shows how many times the argument has been multiplied to itself to get the value of base. Keep this definition in mind and use the logarithmic form i.e. ${\log _a}m = n$ , where $a$ denotes the argument and $m$ denotes the base. The logarithm form ${\log _a}m = n$ means that the value is ${a^n} = m$ .

Complete step-by-step answer :
The logarithmic form of ${a^n} = m$ is ${\log _a}m = n$ , where in ${\log _a}m = n$ ; $a$ , denotes the argument and $m$ denotes the base.
We have been given that ${5^3} = 125$ .
After comparing ${5^3} = 125$ with ${a^n} = m$ , we get, $a = 5,m = 125,n = 3$
Now, we can substitute the value of $a = 5,m = 125,n = 3$ in ${\log _a}m = n$ to find the logarithmic form of ${5^3} = 125$ .
$\Rightarrow {\log _a}m = n\\\ \Rightarrow {\log _5}125 = 3$
Therefore, the logarithmic form of ${5^3} = 125$ is ${\log _5}125 = 3$

Additional Information :
The concept of logarithm was introduced by John Napier in the year 1614. They were used to simplify calculations with the help of a log table that converts complex operations into simple addition and subtraction. Logarithms are widely adopted over the centuries by many scientists, professors, researchers, surveyors, navigators, engineers etc.
Logarithm is a mathematical operation. It helps to find out how many times a number i.e. base is multiplied to itself in order to reach a specific number.
The logarithms whose base is $e$ are written in the form of natural log i.e. $\ln {\rm{ x}}$ .
When logarithms have base 10, then the base is not shown. Symbolically, the logarithm of $a$ to the base 10 is given by $\log {\rm{ a}}$ .
Some of the important logarithm rules are as follows:
$\log \left( {ab} \right) = \log \left( a \right) + \log \left( b \right)$
The logarithm of the product of two numbers is given by the sum of their individual logarithmic values.
\log \left( {{\raise0.7ex\hbox{ a } \\!{\left/ {\vphantom {a b}}\right.} \\!\lower0.7ex\hbox{ b }}} \right) = \log \left( a \right) - \log \left( b \right)
The logarithm of the division of two numbers is equal to the difference between the logarithms of the individual numbers.

Note : Keep this in mind while solving the question that the value of argument and base cannot be interchanged. Students often get confused where to place arguments and where to place the base in the logarithmic form.
If ${a^m} = n$ , then ${\log _n}a \ne m$ .
Also, if ${a^m} = n$ , then ${\rm{lo}}{{\rm{g}}_n}m \ne a$ .