A logarithm can be described in the simplest terms to answer the question “how many times a number is multiplied by itself to get a certain number?”
For example, how many 3s do we multiply to get 27? The answer is calculated by 3 × 3 × 3 = 27. So, three had to be multiplied by itself three times in order to get 27.
The writing of logs is done in a certain way. In the above example, for instance, the log is written as follows:
The number of threes that are required to get 27 is 3. Therefore, it is written as:
\(\log_3 27 = 3\)
Another example: How many 2s are multiplied to get 16?
Answer: 2 × 2 × 2 × 2 = 16. So, four 2s had to be multiplied in order to get 16. Therefore, the logarithm is 4. This can be written in the form, \(\log_2 16 = 4\). This is the reason why the expressions 2 × 2 × 2 × 2 = 16 and \(\log_2 16 = 4\) are said to be the same.
The number that is multiplied is referred to as the base. In the case above, the base is 2. So, we can say:
If the numbers m, x, and n are related as:
\(m^x = n\)
Then \(x\) is said to be the logarithm of the number n to the base m and is written as:
\(\log_m n = x\)
It is important to note that there are three numbers at play here:
Therefore, the logarithm of a number is the value of the index. Examples:
|
43 = 64 |
Log of 64 to the base of 4 is 3 |
\(\log_4 64 = 3\) |
|
5-3 = \(\frac{1}{125}\) |
Log of \(\frac{1}{125}\) to the base 5 = -3 |
\(\log_5 \frac{1}{125} = -3\) |
|
a0 = 1 |
Log of 1 to the base a = 0 |
\(\log_0 1 = a\) |
|
a1 = a |
Log a to the base of a is 1 |
\(\log_a a = 1\) |
The following are more examples of the same:
Example 1. What is the answer to \(\log_5 625\)?
Solution: The question is asking the number of 5s that are needed to be multiplied in order to get 625. The number of 5s is 4. This is because, if you multiply four 5s you get 625. That is, 5 x 5 x 5 x 5 = 625. Therefore, the answer can be written as:
Answer: \(\log_5 625 = 4\)
Example 2. What is the answer to \(\log_2 64\)?
Solution: The question is asking for the number of 2s that are needed to be multiplied in order to get 64. The number of 2 that are multiplied in order to get 64 is 6. This is because, if you multiply six 2s, you get 64. That is, 2 x 2 x 2 x 2 x 2 x 2 = 64. Therefore, the answer can be written as:
Answer: \(\log_2 64 =6\)
Please note if a logarithm is written without a base, consider the base as ‘10’
\(\log_{10}1000 = 3\)
Log value can be negative, look at the below example
\(\log_{10}0.1 = -1\)
why? Because this means \(10^{-1} =0.1\)
\(\log_50.008 = -3 \text{ as } 5^{-3} = \frac{1}{5^3} = 0.008\)
If \(\log_zn = \log_zm = x \textrm{ then } z^x = n \textrm{ and } z^x = m\)
\(\therefore \log_zn = \log_zm \)
⇒ \(n = m\)
