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I was studying logarithms, and had to solve the problem:
If $\log 8 = 0.90$, find $\log 0.125$.
I found out the answer to be $-0.90$. That was easy. But my text book has given the answer as:
$$-0.90 = \bar{1}.10$$
Now what that bar above $1$ is for ? My understanding is that it is used to signify a repeating digit. But how can $-0.90 = \bar{1}.10$ ?
Can someone please explain this to me.
Thank you!
This was a very common notation when decimal logarithms were involved. The starting point is that if $$ M/N=10^k $$ for two positive numbers $M$ and $N$, then their decimal logarithms have the same mantissa (fractional part) and we can write $$ \log_{10} M= k + \log_{10} N $$ In particular, any number $M>0$ can be uniquely written as $$ M=10^k N $$ with integer $k$ and $1\le N<10$. So the logarithmic tables were compiled, say, for numbers from $1$ to $1000$, but only showed the mantissa; when the logarithm of, say, $10.4$ was needed, one looked at $104$ finding the mantissa to be $01703$ and so could conclude $\log_{10}10.4=1.01703$. If one knows the decimal logarithm of $2$ $$ \log_{10}2=0.30103 $$ (equality is meant as “approximate”), then $\log_{10}0.02=-2+0.30103$, which was written as $$ \log_{10}0.02=\overline{2}.30103 $$ to ease lookup in logarithmic tables, because, as already said, only mantissas were shown.
In your case, $0.125=1/8$, so $$ \log_{10}0.125=-\log_{10}8 = -3\log_{10}2=-0.90309=-1+0.09691=\overline{1}.09691 $$ (use the “complement to $9$” rule for writing the mantissa).
