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  4. Number of zeros after decimal before a significant figure in (75)-10 is (use log 10 2=0.301 and log 10 3=0.477 )

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Q. Number of zeros after decimal before a significant figure in $(75)^{-10}$ is (use $\log _{10} 2=0.301$ and $\log _{10} 3=0.477$ )

Continuity and Differentiability

Solution:

$N =(75)^{-10}$
$\log _{10} N =-10 \log (75) $
$=-10\left[\log _{10} 25+\log _{10} 3\right] $
$=-10\left[2 \log _{10} 5+\log _{10} 3\right] $
$=-10\left[2\left\{\log _{10} 10-\log _{10} 2\right\}+\log _{10} 3\right] $
$=-10[2\{1-0.301\}+0.477] $
$=-10[0.699 \times 2+0.477] $
$=-10[1.398+0.477] $
$=-10[1.875] \Rightarrow \text { characteristic of } N =-19= p $
$=-18.75 \Rightarrow \text { Number of zeros after decimal }=| p |-1=|-19|-1=19-1=18$