Question

Between what two integers does log 2 1000 lie

Answer 1

Solution:

Given;

$\log_21000$

From law of logarithm;

$\log_ba=\frac{\log_{10}a}{\log_{10}b}$

Thus;

$\begin{gathered} \log_21000=\frac{\log_{10}1000}{\log_{10}2} \\ \operatorname{\log}_21000=\frac{\operatorname{\log}_{10}10^3}{\operatorname{\log}_{10}2} \\ \end{gathered}$

Another law of logarithm;

$\log_ba^c=c\log_ba$

Thus;

$\begin{gathered} \log_21000=\frac{3\log_{10}10}{\log_{10}2} \\ \log_aa=1 \\ \log_21000=\frac{3\left(1\right)}{\log_{10}2} \\ From\text{ a table of common logarithm;} \\ \log_{10}2=0.3010 \\ \log_21000=\frac{3}{0.301} \\ \log_21000=\frac{3000}{301} \\ \log_21000=9\text{ remainder 291} \\ \end{gathered}$

FINAL ANSWER

$\log_21000\text{ lies between }9\text{ and }10$