Natural Logarithm Series

In mathematics, the natural logarithm is main division. Now we are going to explain about natural logarithm series. Basic natural logarithm is defined as function of ln(x) = log_e^x, in logarithmic series there is no series about natural logarithm function like ln(x), but there is simple series in ln (1+x).

That is ln (1+x) = (x- x^2/2 + x^3/3 - x^4/4 +..)

Here are two things that are sign change to plus and minus alternating for the logarithm series.

Now we are going see about natural logarithm series.

Some basic natural logarithm series:

Natural log series:

1. ln(x) = x-1/x + 1/2(x-1/x)^2 + 1/3(x-1/x)^3 +..

2. ln(1+x) = x - x^2 + x^3/3 - x^4/4 +..

3. ln (1/x) = (1-x) + 1/2(1-x)^2 + 1/3(1-x)^3+..

4. ln (1+x) = x - 1/2x^2 + 1/3x^3 -..

5. ln(1+n/1-n) = 2 (n + n^3/3 + n^5/5+..+ n^2n-1/2n-1 +..)

6. ln (x+1) - ln (x-1) = 2(1/x + 1/3x^2 + 1/5x^5 + ..)

These are the basic important logarithm series. But exponential and logarithm series are there.

Some problems about natural logarithm series:

Problem 1:

To expand the natural logarithm function of ln ((1+x)/(1+2x))

Solution:

Given function is     ln ((1+x)/(1+2x))

We would rewrite the function depended upon logarithm series.

Like,    ln((1+n)/(1-n)) = ln(1+n) - ln (1-n)

Now the function is

ln((1+x)/(1+2x)) = ln(1+x) - ln(1+2x)

Here just taking the series,

 = (x - x^2/2 + x^3/3 - x^4/4 + ..) - (2x - (2x)^2/2 + (2x)^3/3 - (2x)^4/4 + ..)

 = (x - x^2/2 + x^3 /3- x^4/4 +..) -2x + (4x^2)/2 - (8x^3)/3 + (16x^4)/4 -..

 = -x + (3x^2)/2 - (7x^3)/3 + (15x^4)/4 - ..

Now      ln((1+x)/(1+2x)) = -x + (3x^2)/2 - (7x^3)/3 + (15x^4)/4 - ..


Problem 2:

To expand the logarithm function of ln(1+x)^5.

Solution:

Given function is ln (1+x) ^5.

Here we can apply logarithmic law, that is loga^x = x loga.

Now the function is   ln(1+x)^5 = 5log(1+x).

Here just taking the series,

 = 5 (x - x^2/2 + x^3/3 - x^4/4 +..)

Now                         ln(1+x)^5 = 5 (x - x^2/2 + x^3/3 - x^4/4 +..)

Some practices problems for understanding about natural logarithm series:

Problem 1:

To expand the logarithm function of ln(1+x)/(1+3x) using the logarithm series.

Answer = -2x +8x^2/2 - 26x^3/3 +..

Problem 2:

To expand the logarithm function of ln(1+x)^-3 using the logarithm series.

Answer = 3(-x + x^2/2 -x^3/3 +..)

Problem 3:

To expand the logarithm function of ln(1+x)^10  using the logarithm series.

Answer = 10(x - x^2/2 +x^3/3 -..)

That’s all about natural logarithm series.