Question

If the value of \[\log 3=0.477\] and \[{{\left( 1000 \right)}^{x}}=3\], then value of x will be,
(a) 0.159
(b) 0.62
(c) 0.162
(d) 0.59

Answer 1

Hint: Use logarithm of a number in its exponential form. Compare the given expression to the general expression \[x={{b}^{y}}\]and simplify the expression. Substitute the value of \[\log 3\].

Complete step-by-step answer:
We have been given that, \[\log 3=0.477\].
We are also given that, \[{{\left( 1000 \right)}^{x}}=3\]. We need to find the value of x.
An expression that represents repeated multiplication of the same factor is called a power, for eg: \[{{5}^{2}}\].
The number 5 is called the base and the number 2 is called the exponent. The exponent corresponds to the number of times the base is used as a factor.
Here we are given, \[{{\left( 1000 \right)}^{x}}=3\].
The logarithm of a number is the exponent by which another fixed value, the base has to be raised to produce that number.
Thus, \[{{\left( 1000 \right)}^{x}}=3\] is similar to the general expression, \[x={{b}^{y}}\].
Where, y is the logarithm base b of x.
\[\begin{align}
  & \therefore x={{b}^{y}} \\
 & y={{\log }_{b}}\left( x \right) \\
\end{align}\]
Similarly, \[{{\log }_{a}}\left( {{x}^{p}} \right)=p\left( \log a \right)x\].
\[{{\left( 1000 \right)}^{x}}=3\]
If we are taking log to the base 10 on both sides, we get
\[{{\log }_{10}}{{1000}^{x}}={{\log }_{10}}3\].
We know that 1000 can be written as \[{{10}^{3}}\].
\[{{\log }_{10}}{{10}^{3x}}={{\log }_{10}}3\]
Thus we can write that, \[3x={{\log }_{10}}3\].
\[\therefore x=\dfrac{{{\log }_{10}}3}{3}=\dfrac{0.477}{3}=0.159\].
Hence, we got the value of x = 0.159.
\[\therefore \]Option (a) is the correct answer.

Note: It is given that, \[{{\left( 1000 \right)}^{x}}=3\]. We found the value of x = 0.159. Now if we want to check the answer we got, just substitute x = 0.159.
\[\therefore {{\left( 1000 \right)}^{0.159}}=2.99\], which is approximately equal to the value 3.
\[\therefore {{\left( 1000 \right)}^{0.159}}=3\]