I have a very simple question. I am confused about the interpretation of log differences. Here a simple example: $$\log(2)-\log(1)=.3010$$ With my present understanding, I would interpret the result as follows: the number $2$ is $30,10\%$ greater than $1,$ which is obviously false. Can anyone lead me to the right interpretation?
Thanks.
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$\begingroup$Rather, it means that $2$ is (approximately) $10^{0.3010}$ times $1$.
More generally, for any positive $x,y,$ we have $$\log(x)+\log(y)=\log(xy),$$ and for any positive $c,$ we have (assuming we're dealing with base-$10$ logarithms rather than natural logarithms) $$c=\log(10^c).$$ Hence, the following are equivalent: $$\log(a)-\log(b)=c\\\log(a)=c+\log(b)\\\log(a)=\log(10^c)+\log(b)\\\log(a)=\log(10^cb).$$ By the uniqueness property of logarithms, this is then equivalent to $a=10^cb.$ If dealing with natural logarithms instead, $\log(a)-\log(b)=c$ is equivalent to $a=e^cb.$
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