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
Dec 14, 18 04:05 PM
Here is how to convert square feet to acres - To convert square feet to ...
New math lessons
Your email is safe with us. We will only use it to inform you about new math lessons.
Dec 14, 18 04:05 PM
Here is how to convert square feet to acres - To convert square feet to ...
Our Top Pages
Formula for percentage
Compatible numbers
Basic math test
Basic math formulas
Types of angles
Math problem solver
Algebra word problems
Surface area of a cube
Finding the average
Scale drawings
Everything you need to prepare for an important exam!
K-12 tests, GED math test, basic math tests, geometry tests, algebra tests.
Real Life Math Skills
Learn about investing money, budgeting your money, paying taxes, mortgage loans, and even the math involved in playing baseball.