ln(x): calculation and implementation in different programming languages

The natural logarithm of a number is that number’s logarithm to the base of e (a mathematical constant). Shown as a mathematical statement:

The natural logarithm of a number is that number’s logarithm to the base of e

To show this better, we could also use:

Now, we go back to our first problem: “How to calculate ln(x)?”

For the first step, we are going to find the Maclaurin’s series in order to find our Answer:

So if -1≤t≤1:

calculation of ln(1+x)

And for the final step:

calculation of ln(1+x)

We can also show this equation by:

calculation of ln(1+x)

In order to calculate ln(x) itself, we use a simple trick: setting the value of (x) to (t-1):

As conclusion:

But if you remember, we said that this equation only works if -1≤x≤1.

So how can we rewrite this formula to calculate ln(x), for x ∈ ℝ?

But first, let’s look into some basic rules of natural logarithm:

Now back to our own problem, we can show each number as a scientific notation. Shown as a mathematical expression:

we can show each number as a scientific notation, ‘a’ is called “Coefficient” and ‘b’ is called “Exponent”.

For example:

Now, let’s modify this scientific notation:

Since -10<a<10, then -1<0.1a<1. As said before, our ln(x) equation worked only if -1≤x≤1. Then:

since 10 isn’t less than 1:

One quick look at what we got:

Now let’s calculate each part with the expression we had before:

calculation of ln(0.1a)
calculation of ln(0.1)

Concluding:

the final equation

Now that we have the final equation, let’s get into implementing it in Node.js and Python 3.8.

Implementation in Python 3.8

Implementation of the ln function in Python is shown below. First we convert x to its scientific notation, then we do the rest of calculations. Note that since we can’t calculate this to infinity, we have to end our calculations at some point; So we set our limit to endOfCalc. Obviously, the larger endOfCalc is, the more accurate our result will be. For example, I set endOfCalc to 370 because it still runs smoothly and gives us the result with the accuracy of 10 digits after the decimal point (at most). Take a look:

endOfCalc = 370
def ln(x):
scientific_notation = "{:.12e}".format(x)
scientific_notation = scientific_notation.replace('+','')
a, b = map(float, scientific_notation.split('e'))
output = 0
for i in range(1,endOfCalc):
output += (((a/10)-1)**i)*((-1)**(i+1))/i
output -= (b+1)*((-0.9)**i)*((-1)**(i+1))/i
return round(output,10)

Now to test it, we add ln(e) (e is approximately equal to 2.718281828459) to the program:

print(ln(2.718281828459))

When you run it, you’ll get 1.0; Means that everything worked out just fine.

A wise thing to do in order to reduce the execution time, is to pre-calculate ln(0.1), which is approximately equal to -2.30258509299. So the final code is:

endOfCalc = 370
ln0_1 = -2.30258509299;
def ln(x):
scientific_notation = "{:.12e}".format(x)
scientific_notation = scientific_notation.replace('+','')
a, b = map(float, scientific_notation.split('e'))
output = 0
for i in range(1,endOfCalc):
output += (((a/10)-1)**i)*((-1)**(i+1))/i
return round(output-(b+1)*ln0_1,10)

Again, you can test it with the method explained before.

Implementation in Node.js

Implementation of the ln function in Node.js is shown below. First we convert x to its scientific notation, then we do the rest of calculations. Note that since we can’t calculate this to infinity, we have to end our calculations at some point; So we set our limit to endOfCalc. Obviously, the larger endOfCalc is, the more accurate our result will be. For example, I set endOfCalc to 370 because it still runs smoothly and gives us the result with the accuracy of 10 digits after the decimal point. Take a look:

endOfCalc = 370;
function ln(x){
x = x.toExponential();
x = x .replace('+','');
let listed = x.split("e");
a = Number(listed[0]);b = Number(listed[1]);
output = 0;
for(i = 1;i < endOfCalc;i++){
output += (((a/10)-1)**i)*((-1)**(i+1))/i;
output -= (b+1)*((-0.9)**i)*((-1)**(i+1))/i;
}
return (output).toFixed((10));
}

Now to test it, we add ln(e) (e is approximately equal to 2.718281828459) to the program:

console.log(ln(2.718281828459))

When you run it, you’ll get 1.0000000000; Means that everything worked out just fine.

A wise thing to do in order to reduce the execution time, is to pre-calculate ln(0.1), which is approximately equal to -2.30258509299. So the final code is:

endOfCalc = 370;
ln0_1 = -2.30258509299;
function ln(x){
x = x.toExponential();
x = x .replace('+','');
let listed = x.split("e");
a = Number(listed[0]);b = Number(listed[1]);
output = 0;
for(i = 1;i < endOfCalc;i++){
output+=(((a/10)-1)**i)*((-1)**(i+1))/i;
}
return (output-(b+1)*ln0_1).toFixed((10));
}

Again, you can test it with the method explained before.

Shahid Ezheie 1 Highschool student, Iran