# Properties of logarithms

The table below will help you understand the properties of logarithms quickly.

Notice that log x  = log10 x

If you do not see the base next to log, it always means that the base is 10. We have also rounded to the nearest thousandth.

 x 1 2 3 4 5 6 7 8 9 10 15 20 log x 0 0.301 0.477 0.602 0.699 0.778 0.845 0.903 0.954 1 1.176 1.301

Now, what do you notice about the following pairs of statements?

log 2 + log 3 and log (2 x 3)

log 4 + log 5 and log (4 x 5)

Here is what we notice about them.

log 2 + log 3 = 0.301 + 0.477 = 0.778 and 0.778 = log (6) = log (2 x 3)

Therefore, log 2 + log 3 = log (2 x 3)

By the same token,

log 4 + log 5 = 0.602 + 0.699 = 1.301 and 1.301 = log (20) = log (4 x 5)

Therefore, log 4 + log 5 = log (4 x 5)

In general, for any positive numbers, M, N and b, where b is not equal to 1, we have the following product property

logb MN = logb M + logb N

Second, what do you notice about the following pairs of statements?

log (10 / 5) and log 10 - log 5

log (8 / 2)  and log 8 - log 2

Here is what we notice about them.

log 8 - log 2 = 0.903 - 0.301 = 0.602 and 0.602 = log (4) = log (8 / 2)

Therefore, log 8 - log 2 = log (8 / 2)

By the same token,

log 10 - log 5 = 1 - 0.699 = 0.301 and 0.301 = log (2) = log (10 / 5)

Therefore, log 10 - log 5 = log (10 / 5)

In general, for any positive numbers, M, N and b, where b is not equal to 1, we have the following quotient property

logb (M / N) = logb M - logb N

Finally, what do you notice about the following pairs of statements?

log (23) and 3 x log 2

log (32) and 2 x log 3

Here is what we notice about them.

log (23) = log 8 = 0.903

3 x log 2 = 3 x 0.301 = 0.903

Therefore, log (23) = 3 x log 2

log (32) = log 9 = 0.954

2 x log 3 = 3 x 0.477 = 0.954

log (32) = 2 x log 3

In general, for any positive numbers, M, N and b, where b is not equal to 1, we have the following quotient property

logb (Mx) = x logb M

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