Find the number of zeros after decimal point in $0.2^{25}$

Question

Find the number of zeroes immediately after decimal point in $(0.2)^{25}$,given that $\log 2=0.30101$

My attempt: I found the answer as $17.\dots$

Should we add $1$ as $17$ is the characteristic or leave it as 17?

Answer 1

You've found that $0.2^{25} = 10^{-17.47\dots}$, so $ 10^{-18} < 0.2^{25} < 10^{-17}, $ that is $$ \underbrace{0.0\dots 0}_\text{$18$ zéros}1 < 0.2^{25} < \underbrace{0.0\dots 0}_\text{$17$ zéros}1, $$ so we conclude that there are $17$ zéros after the decimal point.

Answer 2

We have $$ \log_{10} 0.2^{25} = 25 \cdot \log_{10} \frac 2{10} = 25 \cdot (\log_{10} 2 - 1) \approx -17.4743 $$ That gives, by monotinicity of $x \mapsto 10^{x}$ that $$ 10^{-18} < 10^{-17.4743\ldots} = 0.2^{25} < 10^{-17}, $$ hence $0.2^{25}$ has 17 zeros after the decimal point, no need to add 1.

Answer 3

let $(0.2)^{25}=x$

$\implies \log (0.2)^{25}=\log x $

$\implies 25\log (0.2)=\log x $

$\implies 25(\log 2-\log 10=\log x) $

$\implies 25(0.30101-1)=\log x $

$\implies \log x = -17.47475$

$\therefore$Characteristic=17=number of zeroes immediately after decimal point in $(0.2)^{25}$

Note- "$characteristic+1$" gives total number of digits.