Question

Solve the value of the given expression, $-$$\log \left( 4.6\times {{10}^{-5}} \right)$.

Answer 1

Hint: To solve this problem we should first know about some logarithmic properties, i.e.
In the logarithms $\log \left( a\times b \right)=\log \left( a \right)+\log \left( b \right)$ and also
$n{{\log }_{a}}b={{\log }_{a}}{{b}^{n}}$, first we will apply the property $\log \left( a\times b \right)=\log \left( a \right)+\log \left( b \right)$ then after that we will apply second mentioned property and from that we can find the value of the given expression and also to solve this problem we need to know that $\log \left( 46 \right)=1.66$.

Complete step by step answer:
We are given the expression as,
$-$$\log \left( 4.6\times {{10}^{-5}} \right)$
And we have to find it’s value,
So we will solve the this as,
$=-\log \left( 4.6\times {{10}^{-5}} \right)$
$=-\log \left( 46\times {{10}^{-6}} \right)$
And we know that $\log \left( a\times b \right)=\log \left( a \right)+\log \left( b \right)$ so by applying this property on the above expression we get,
\[=-\left[ \log \left( 46 \right)+\log \left( {{10}^{-6}} \right) \right]\]
We also know that $n{{\log }_{a}}b={{\log }_{a}}{{b}^{n}}$, applying this property on the term $\log \left( {{10}^{-6}} \right)$ in above expression, we get
\[=-\left[ \log \left( 46 \right)-6\log \left( 10 \right) \right]\]
Putting log(10) = 1, we get
$=-[\log \left( 46 \right)-6]$
And also We know that the value of log(46) = 1.66, so
$\begin{align}
  & =-[1.66-6] \\
 & =4.34 \\
\end{align}$

Hence, the value of the given expression -$\log \left( 4.6\times {{10}^{-5}} \right)$ we get as 4.34

Note: You need to read all the properties of the logarithms in order to solve these kinds of problems and try to practice more problems of this type so that you can easily apply the logarithmic properties. If you don’t know these properties like $\log \left( a\times b \right)=\log \left( a \right)+\log \left( b \right)$ then you would get stuck up in the given question and will not be able to solve it so these properties are very important and you should remember both properties $\log \left( a\times b \right)=\log \left( a \right)+\log \left( b \right)$ and $n{{\log }_{a}}b={{\log }_{a}}{{b}^{n}}$ to solve problems of this kind in future.