Question

How do you rewrite \[{{\log }_{11}}1331=3\] in exponential form ?

Answer 1

Hint: In the given question, we have been asked to solve \[{{\log }_{11}}1331=3\]. In order to write the expression into exponential form, first we apply the property of logarithm which states that the definition of logarithm i.e. using the definition of log, If \[x\] and b are positive real numbers and b is not equal to 1, then \[{{\log }_{b}}\left( x \right)=y\] is equivalent to \[{{b}^{y}}=x\]. Then to eliminate log function we need to convert logarithmic equation into the exponential equation as logarithmic functions are the inverse of the exponential functions and simplify the expression further.

Formula used:
\[\log {{\left( x \right)}^{a}}=a\log x\]
Using the definition of log, If \[x\] and b are positive real numbers and b is not equal to 1,
Then \[{{\log }_{b}}\left( x \right)=y\] is equivalent to\[{{b}^{y}}=x\].

Complete step by step answer:
We have given that, \[{{\log }_{11}}1331=3\]
As, we know that
\[1331=11\times 11\times 11={{11}^{3}}\]
Therefore,
\[{{\log }_{11}}{{\left( 11 \right)}^{3}}=3\]
Taking the LHS,
Using the property of logarithm which states that \[\log {{\left( x \right)}^{a}}=a\log x\]
Applying this property in the above expression, we get
\[3\times {{\log }_{11}}\left( 11 \right)\]
As we know that,
\[{{\log }_{11}}11=1\]
Using the definition of log.If \[x\] and b are positive real numbers and b is not equal to 1.Then \[{{\log }_{b}}\left( x \right)=y\] is equivalent to \[{{b}^{y}}=x\]. Applying this property in the given expression, we obtain
\[{{11}^{1}}=11\]
Substitute the value of \[{{\log }_{11}}11=1\] in the above solved expression of LHS, we get
\[\Rightarrow 3\times {{\log }_{11}}\left( 11 \right)\]
\[\therefore 3\times 1=3=RHS\]

Hence, the exponential form of \[{{\log }_{11}}1331=3\] is equals to \[{{3}^{1}}=3\].

Note:To solve these types of questions, we used the basic formulas of logarithm. Students should always require to keep in mind all the formulae for solving the question easily. After applying log formulae to the equation, we need to solve the equation in the way we solve general linear equations. Logarithmic functions are the inverse of exponential functions with the same bases i.e. if \[x\] and b are positive real numbers and b is not equal to 1, then \[{{\log }_{b}}\left( x \right)=y\] is equivalent to \[{{b}^{y}}=x\].