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Taking the antilog of a number undoes the operation of taking the log. Therefore, since Log₁₀(1000) = 3, the antilog₁₀ of 3 is 1,000. Taking the antilog of a number simply raises the base of the logarithm in question to that number.

Logs and Proportional Change

A series of numbers that increases proportionally will increase in equal amounts when converted to logs. For example, the numbers in the first column of Table 1 increase by a factor of 1.5 so that each row is 1.5 times as high as the preceding row. The Log₁₀ transformed numbers increase in equal steps of 0.176.

Table 1. Proportional raw changes are equal in log units.

As another example, if one student increased their score from 100 to 200 while a second student increased their's from 150 to 300, the percentage change (100%) is the same for both students. The log difference is also the same, as shown below.

$Log_{10}(100)=2.000$
$Log_{10}(200)=2.301$
$Difference:0.301$

$Log_{10}(150)=2.176$
$Log_{10}(300)=2.477$
$Difference:0.301$

Arithmetic Operations
Rules for logs of products and quotients are shown below.
$Log(AB)=Log(A)+Log(B)$
$Log(A/B)=Log(A)-Log(B)$

For example,

$Log_{10}(10\times 100)=Log_{10}(10)+Log_{10}(100)=1+2=3.$

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