Question

How do you evaluate \[{{\log }_{10}}\left( \dfrac{1}{{{10}^{x}}} \right)\]?

Answer 1

Hint: This question can be solved by using one of the properties for log which is \[\log \left( \dfrac{x}{y} \right)=\log x-\log y\]. Here, x will be 1 and y will be \[{{10}^{x}}\]. Then you should use another property which is \[\log \left( {{a}^{x}} \right)=x\log \left( a \right)\]. Then you will get your answer.

Complete step by step solution:
According to the problem, we are asked to evaluate the equation \[{{\log }_{10}}\left( \dfrac{1}{{{10}^{x}}} \right)\]--- ( 1 )
Therefore, by using the property of log that is \[\log \left( \dfrac{x}{y} \right)=\log x-\log y\], in equation 1, we get:
Here , x = 1 and y = \[{{10}^{x}}\].
Therefore, we can get:
\[\Rightarrow {{\log }_{10}}\left( \dfrac{1}{{{10}^{x}}} \right)={{\log }_{10}}\left( 1 \right)-{{\log }_{10}}\left( {{10}^{x}} \right)\] ---- (2)
We also know that log(1) = 0. ---- (3)
And for \[\log \left( {{10}^{x}} \right)\], we use another property of log, which is \[\log \left( {{a}^{x}} \right)=x\log \left( a \right)\]. Here, x is x and a = 10. Therefore, by using the property we get,
\[\Rightarrow {{\log }_{10}}\left( {{10}^{x}} \right)=x{{\log }_{10}}\left( 10 \right)\] ------ (4)
By substituting equation 3 and equation 4 in equation 2, we get
\[\begin{align}
  & \Rightarrow {{\log }_{10}}\left( \dfrac{1}{{{10}^{x}}} \right)={{\log }_{10}}\left( 1 \right)-{{\log }_{10}}\left( {{10}^{x}} \right) \\
 & \Rightarrow {{\log }_{10}}\left( \dfrac{1}{{{10}^{x}}} \right)=0-x{{\log }_{10}}\left( 10 \right) \\
\end{align}\]
\[\Rightarrow {{\log }_{10}}\left( \dfrac{1}{{{10}^{x}}} \right)=-x{{\log }_{10}}\left( 10 \right)\] ------ (5)
But, we also know that \[{{\log }_{10}}\left( 10 \right)=1\]. If we take this as equation (6)
\[\Rightarrow {{\log }_{10}}\left( 10 \right)=1\] ------ (6)
Finally, by substituting equation 6 in equation 5, we get :
\[\Rightarrow {{\log }_{10}}\left( \dfrac{1}{{{10}^{x}}} \right)=-x\left( 1 \right)\]
\[\Rightarrow {{\log }_{10}}\left( \dfrac{1}{{{10}^{x}}} \right)=-x\]
So, we have found that after evaluating the given equation \[{{\log }_{10}}\left( \dfrac{1}{{{10}^{x}}} \right)\] we get the answer as -x .
Therefore, the solution of the given equation \[{{\log }_{10}}\left( \dfrac{1}{{{10}^{x}}} \right)\] is \[{{\log }_{10}}\left( \dfrac{1}{{{10}^{x}}} \right)=-x\].

Note: You should always remember these properties of log. Without knowing these, you cannot solve these kinds of problems. So it is very important to learn them. Also, you should know that \[{{\log }_{a}}\left( {{a}^{x}} \right)\] is always x for all a > 0 except for a = 1. From here, we get \[{{a}^{{{\log }_{a}}\left( x \right)}}=x\] . If you know this, the above question can be solved very easily.