Question

Determine how many zeros are there between the decimal point and the first significant digit in ${\left( {\dfrac{1}{2}} \right)^{1000}}$, if ${\log _{10}}2 = 0.30103$.

Answer 1

Hint: Here, we will simplify the given solution using the properties of logarithms.

Complete step-by-step answer:
We are given the value as,
$y = {\left( {\dfrac{1}{2}} \right)^{1000}}$
Now we will use the identity
$\log {a^m} = m\log a$
We will get,
$\log y = 1000\log \dfrac{1}{2}$
Now we will use the identity
$\log \dfrac{m}{n} = \log m - \log n$
We will get,
$
  \log y = 1000(\log 1 - \log 2) \\
  \log y = 1000(0 - 0.30103) \\
  \log y = - 301.103 \\
$
The negative sign implies that the zeros are present after the decimal point.
Therefore, there are 301 zeros between the decimal point and the first significant digit in ${\left( {\dfrac{1}{2}} \right)^{1000}}$.

Note: In these types of questions, we should use the properties of logarithmic functions to simplify and arrive at a solution. Logarithmic functions are often used to simplify the complex expressions.