Question

How do you evaluate ${\log _7}\left( {\dfrac{1}{{49}}} \right)$ ?

Answer 1

Hint: This equation can be easily solved by using basic formula of logarithm. Simplify the equation by using the property of logarithm, we know that
${\log _a}{\left( b \right)^m} = m{\log _a}\left( b \right)$

Complete step by step answer:
According to question
Given that, ${\log _7}\left( {\dfrac{1}{{49}}} \right)$
On simplifying we get
$ \Rightarrow {\log _7}{\left( {49} \right)^{ - 1}}$
$ \Rightarrow {\log _7}{\left( {{7^2}} \right)^{ - 1}}$
On using the property of log we get
  ${\left( {{a^m}} \right)^n} = {\left( a \right)^{mn}}$
$ \Rightarrow {\log _7}{\left( 7 \right)^{ - 2}}$
Again using the logarithm property,
${\log _a}{\left( b \right)^m} = m{\log _a}\left( b \right)$
$ \Rightarrow - 2{\log _7}\left( 7 \right)$
Using the logarithm property,
${\log _a}a = 1$
On further solving we get ,
${\log _7}\left( {\dfrac{1}{{49}}} \right)$$ = - 2$

Hence the value of ${\log _7}\left( {\dfrac{1}{{49}}} \right)$ is -2.

Note: A logarithm is the power to which a number must be raised in order to get some other number. For example, the base ten logarithm of 100 is 2, because ten raised to the power of two is 100 that is log 100 = 2.