Question

If it is given ${{\log }_{10}}7=a$, then find the value of ${{\log }_{10}}\left( \dfrac{1}{70} \right)$.
(a) \[\left( 1+a \right)\],
(b) ${{\left( 1+a \right)}^{-1}}$,
(c) $\dfrac{a}{10}$,
(d) $\dfrac{1}{10a}$.

Answer 1

Hint: We assume the variable for the value of given ${{\log }_{10}}\left( \dfrac{1}{70} \right)$. We use the result ${{\log }_{y}}\left( \dfrac{b}{c} \right)={{\log }_{y}}b-{{\log }_{y}}c$ to proceed through the problem. We then use the result ${{\log }_{y}}\left( a\times b \right)={{\log }_{y}}\left( a \right)+{{\log }_{y}}\left( b \right)$ to proceed further through the problem. We use ${{\log }_{y}}\left( 1 \right)=0$, ${{\log }_{y}}\left( y \right)=1$ and given ${{\log }_{10}}7=a$ to get the required answer.

Complete step by step answer:
Given that we have ${{\log }_{10}}7=a$ and we need to find the value of ${{\log }_{10}}\left( \dfrac{1}{70} \right)$. Let us assume the value of ${{\log }_{10}}\left( \dfrac{1}{70} \right)$ be ‘x’.
$\Rightarrow $ $x={{\log }_{10}}\left( \dfrac{1}{70} \right)$ ---(1).
We know that the value of ${{\log }_{y}}\left( \dfrac{b}{c} \right)={{\log }_{y}}b-{{\log }_{y}}c$, for any given value of y $\left( y>0 \right)$. We use this in equation (1)
$\Rightarrow $ $x={{\log }_{10}}\left( 1 \right)-{{\log }_{10}}\left( 70 \right)$ ---(2).
We know that the value of ${{\log }_{y}}\left( 1 \right)=0$, for any given value of y $\left( y>0 \right)$. We use this in equation (2).
$\Rightarrow $ $x=-{{\log }_{10}}\left( 70 \right)$.
$\Rightarrow $ $x=-{{\log }_{10}}\left( 7\times 10 \right)$ ---(3).
We know that the value of ${{\log }_{y}}\left( a\times b \right)={{\log }_{y}}\left( a \right)+{{\log }_{y}}\left( b \right)$. We use this in equation (3).
$\Rightarrow $ $x=-\left( {{\log }_{10}}\left( 7 \right)+{{\log }_{10}}\left( 10 \right) \right)$ ---(4).
We know that the value of ${{\log }_{y}}\left( y \right)=1$, for any given value of y $\left( y>0 \right)$. And according to the problem, we have given ${{\log }_{10}}7=a$. We use this in equation (4).
$\Rightarrow $ $x=-\left( a+1 \right)$.
$\Rightarrow $ $x=-\left( 1+a \right)$.
∴ We have found the value of ${{\log }_{10}}\left( \dfrac{1}{70} \right)$ as $-\left( 1+a \right)$.

So, the correct answer is “Option A”.

Note: We need to make sure that the base of the logarithmic function should be greater than zero. We can directly make use of formula ${{\log }_{y}}\left( \dfrac{1}{z} \right)=-{{\log }_{y}}\left( z \right)$ in the problem to reduce the calculation time. Similarly, we can expect problems to find the value of ‘a’ as the base is given as 10.