Question

How do you evaluate ${{\log }_{7}}\left( \dfrac{1}{343} \right)$ ?

Answer 1

Hint: First we write $343\;$ in exponential form. $343\;$ should be written in such a way that it is in the powers of $7$. It is easier if we write it in powers of $7$ so that we can cancel the logarithmic base with it. Use the laws of logarithms to remove the exponent and then simplify it.

Complete step by step solution:
The given logarithmic expression is, ${{\log }_{7}}\left( \dfrac{1}{343} \right)$
Firstly, we write $343\;$ in exponential form, as the power of $7$.
Since $343=7\times 7\times 7$
Now, on converting it into exponential form,
$\Rightarrow 343={{7}^{3}}$
Now put it back into a logarithmic expression.
$\Rightarrow {{\log }_{7}}\left( \dfrac{1}{{{7}^{3}}} \right)$
Now let us bring the term in the denominator to the numerator.
Whenever we bring a constant with any power to the numerator, it becomes negative.
In this format.
$\Rightarrow \dfrac{1}{a}={{a}^{-1}}$
Our expression will now become,
${{\log }_{7}}\left( {{7}^{-3}} \right)$
Now, consider the term.
By using the law of powers of logarithms,
If we have the function, $f\left( x \right)={{\log }_{a}}\left( {{b}^{c}} \right)$ .Then we can convert into power form as, $f\left( x \right)=c{{\log }_{a}}\left( b \right)$ .
Here $a=7;b=7;c=-3$
On Substituting,
$\Rightarrow -3{{\log }_{7}}7$
If the base and the logarithm value is the same, they cancel out to get $1$
$\Rightarrow {{\log }_{7}}7=1$
$\Rightarrow \left( -3 \right)\left( 1 \right)$
Now, putting it all together we get,
$\Rightarrow -3$
Hence the given expression is equal to ${{\log }_{7}}\left( \dfrac{1}{343} \right)=-3$ .

Note: One should ensure that the base of the given logarithms is the same before evaluating the expression using the laws of the logarithms. Some of the logarithmic formulae which could be useful are,
Law of powers of logarithms, if we have the function, $f\left( x \right)={{\log }_{a}}\left( {{b}^{c}} \right)$ .Then we can convert into power form as, $f\left( x \right)=c{{\log }_{a}}\left( b \right)$ .
If the base of the logarithm is in fraction form then we use,
Law of base power of logarithms, if we have the expression,$f\left( x \right)={{\log }_{\left( {{b}^{c}} \right)}}a$ .It can also be written as,$f\left( x \right)=\dfrac{1}{c}{{\log }_{b}}c$ .