Logarithmic Series
In mathematics, the logarithmic function is main division. Now we are going to explain about how to help to the students about logarithmic series. Basic logarithmic function is defined as function of `log_b x = ln(x)` , in logarithmic series there is no series about logarithm function like ln(x), but there is simple series in `ln (1+x).`
In mathematics, the logarithmic function is main division. Now we are going to explain about how to help to the students about logarithmic series. Basic logarithmic function is defined as function of `log_b x = ln(x)` , in logarithmic series there is no series about 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 logarithmic series.
Now we are going see about help to the students based on logarithmic series.
If a > 0, by Exponential Theorem
`a^y = 1 + y log a + [y^2 (log a)^2] / 2!+ [y^3 (log a)^3] / 3!` + ............to infinity
putting a = 1 + x
If `a > 0` , by Exponential Theorem
`a^y = 1 + y log a + [y^2 (log a)^2] / 2!+ [y^3 (log a)^3] / 3!+ .` ............... to infinity
putting a = 1 + x
`(1 + x)^y = 1 + y log (1 + x) + [y^2 log (1+x)]^2] / 2! + [y^3 log (1+x)^3] / 3! + ` .. .. ...........................................to infinity.
By Binomial Theorem for any index
` (1 + x)^y = 1 + xy + [ y ( y- 1) x^2] / 2! + [ y ( y - 1) (y - 2) x^3] / 3! + ..` ................... to infinity
Equating these two series
` 1 + y log (1 + x) + [ y^2 log (1 + x)^2 / 2! + ....................... = 1 + xy + [ y (y - 1) x^2] / 2! + [ y (y - 1) (y - 2) x^3] / 3! + .........`
Equating coifficients of y on both sides,
`log (1 + x) = x - x^2/2! + [ (-1) (-2) x^3] / 3! +` ..................... to infinity
`= x - x^2/2 + x^3/3 - x^4/4 +` ...................................to infinity
so
`log (1 + x) = x - x^2 / 2 + x^3 / 3 - x^4 / 4 + ` ...................................to infinity
this series is called Logarithmic Series
Logarithmic series for help to the students online:
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 logarithmic series.
Below are some examples based on logarithmic series
Problem 1: To solve the logarithmic function of` ln ((1+4x)/(1+2x)).`
Solution: Given function is ` ln ((1+4x)/ (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+4x)/(1+2x)) = ln(1+4x) - ln(1+2x)`
Here just taking the series,
` = (4x - (4x)^2/2 + (4x)^3/3 - (4x)^4/4 + ..) - (2x - (2x)^2/2 + (2x)^3/3 - (2x)^4/4 + ..)`
` = (4x -(16x^2)/2 + (64x^3)/3-(256x^4)/4 +..) -2x + (4x^2)/2 - (8x^3)/3 + (16x^4)/4 -..`
` = 2x - (12x^2)/2 + (56x^3)/3 - (240x^4)/4 + ..`
Answer: ` ln((1+4x)/(1+2x)) = 2x - 6x^2 +(56x^3)/3 - (60x^4) + .. `
Problem 2: To solve the logarithmic function of `ln(1+3x)^-3.`
Solution: Given function is `ln (1+3x) ^-3`
Here we can apply logarithmic law, that is` loga^x = x loga.`
Now the function is `ln(1+3x)^-3 = -3log(1+3x).`
Here just taking the series,
`=-3 (3x - 3x^2/2 + 3x^3/3 - 3x^4/4 +..)`
`= -3 (3x - 9x^2/2 + 27x^3/3 - 81x^4/4 +..)`
` = - (9x -(27x^2)/2 + (81x^3)/3 - (243x^4)/4 -..)`
`= -9x +(27x^2)/2 - 27x^3 + (243x^4)/4 +..`
Answer: ` ln(1+3x)^-3 = -9x +(27x^2)/2 - 27x^3 + (243x^4)/4 +..`
Prepare some problems about logarithmic series for help:
Problem 1: To solve the logarithmic function of` ln((1+3x)/(1+2x))` using the logarithmic series.
Answer `= x - (5x^2)/2 + (19x^3)/3 -(65x^4)/4 + ..`
Problem 2: To solve the logarithmic function of ln(1+3x)^-2 using the logarithmic series.
Answer `= -6x + 9x^2 - 18x^3 + 81x^4 + ..`
Problem 3: To solve the logarithmic function of ln(1+x)^-5 using the logarithmic series.
Answer `= -5x + (5x^2)/2 -(5x^3)/3 +..`
