Given an integer N. The task is to find the minimum number of log values needed to calculate all the log values from 1 to N using properties of the logarithm.
Input : N = 6 Output : 3 Value of log1 is already know, i.e. 0. Except this the three log values needed are, log2, log3, log5. Input : N = 4 Output : 2
One of the properties of log function is:
log(x.y) = log(x) + log(y)
Hence, to calculate log(x.y), we must know log values of x and y. Let us understand the concept using an example, for N = 6. Let ans denotes the number of log values needed to find all log values from 1 to 6.
- log(1)=0 (implicit).
- To calculate log(2), we must know its value prior, we can’t find this using property.so, ans become 1.
- To calculate log(3), we must know its value prior, we can’t find this using property.so, ans become 2.
- To calculate log(4), we can use property, log(4)=log(2.2)=log(2)+log(2).As we already find log(2) hence ans remains 2.
- To calculate log(5), we must know its value prior, we can’t find this using property.so, ans become 3.
- To calculate log(6), we can use property, log(6)=log(2.3)=log(2)+log(3).As we already find log(2) and log(3), hence ans remains 3.
The idea is very simple, on observing carefully you will find that you can’t calculate log values of prime number as it has no divisor(other than 1 and itself). So, the task reduces to find all prime numbers from 1 to N.
Below is the implementation of the above approach:
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