How to Calculate Logarithms
The logarithm is denoted as loga b and this entry reads as: The logarithm of x to base b
When calculating logarithms, keep in mind that the numbers a and b must be greater than 0 and a must not be equal to 1.
loga b exists for a> 0, a ≠ 1, b> 0
Logarithms with base a equal to 2, 10 or number e got their names:
loge b whose base is equal to Euler's number e (e = 2.7182818284 ...) is called - Natural logarithm and is denoted ln b .
For example, ln 4 is the same as loge 4, but the ln entry itself says that
the base is e and therefore the notation is shortened.
log10 b whose base is 10 is called Common logarithm and is denoted lg b . For example, lg 6, which is the same as log10 6
log2 b whose base is 2 is called Binary logarithm and is denoted lb b , such logarithms are often used in computer science.
For example, lb 3 is the same as log2 3.
You can easily determine whether the logarithm loga b is negative or positive, for this there is a rule:
if 0 < a > 1 and 0 < b <1 or 0 < a <1 and 0 < b > 1
then the logarithm is negative, otherwise positive
loga b <0 if 0 < a > 1 and 0 < b <1 or 0 < a <1 and 0 < b > 1
For example, these logarithms will be negative log1/3 4, log4 1/3, log2/3 5, log5 2/3, etc.
That is, either a or b must be less than one, but not both.
Calculating the logarithm means finding the exponent to which the number a must be raised to get the number b.
In simple terms, when we calculate the logarithm, we always find the power, and if we raise the number a to this power, we get the number b.
Let's denote x as the unknown power of a, then we can write the following equation: a x = b
Here are some examples:
Given the logarithm of log4 64, we need to find such an exponent that when raising the number 4 to it we should get 64.
Let's write the equation:
4 x = 64 < br />
4 x = 4 3
x = 3
Let's check, raise the number 4 to the power of 3: 4 3 = 64.
In general, any value of the logarithm is always easy to check,
it is enough to raise the number a to a power equal to the value of the logarithm, and if the result is equal to the number b, then the answer is correct.