The table below will help you understand the properties of logarithms quickly.
Notice that log x = log_{10} 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
log_{b} MN = log_{b} M + log_{b} 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
log_{b} (M / N) = log_{b} M - log_{b} N
Finally, what do you notice about the following pairs of statements?
log (2^{3}) and 3 x log 2
log (3^{2}) and 2 x log 3
Here is what we notice about them.
log (2^{3}) = log 8 = 0.903
3 x log 2 = 3 x 0.301 = 0.903
Therefore, log (2^{3}) = 3 x log 2
log (3^{2}) = log 9 = 0.954
2 x log 3 = 3 x 0.477 = 0.954
log (3^{2}) = 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
log_{b} (M^{x}) = x log_{b} M
Feb 17, 19 12:04 PM
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Feb 17, 19 12:04 PM
There is no rational number whose square is 2. An easy to follow proof by contraction.