In this section, you will learn how to find expansion of logarithmic functions.

The series Σ n = 1 to ∞ (−1)^{n+1} x^{n }/n is called a logarithmic series. This series converges for all values of x satisfying |x| < 1. This series converges when x = 1 also.

For all values of x satisfying |x| < 1, the sum of the series is log(1 + x). Thus

log(1 + x) = x − (x^{2}/2) + (x^{3}/3) − (x^{4}/4) + · · ·

for all values of x satisfying |x| < 1.

By taking −x in place of x we get

log(1 − x) = −x − x^{2}/2 − x^{3}/3 − x^{4}/4 − ·· ·

for all values of x satisfying |x| < 1.

Now log [ (1+x)/(1−x)] = log(1+x) − log(1 − x).

Using this we get

log [ (1 + x)/(1 − x)] = 2 [x + x^{3}/3 + x^{5}/5 +............]

Now log [ (1-x)/(1+x)] = log(1-x) − log(1+x).

Using this we get

log [ (1 - x)/(1 + x)] = -2 [x + x^{3}/3 + x^{5}/5 +............]

**Question 1 :**

Write the first 4 terms of the logarithmic series

log (1 + 4x)

**Solution :**

log(1 + x) = x − (x^{2}/2) + (x^{3}/3) − (x^{4}/4) + · · ·

Instead of x, apply 4x

log(1 + 4x) = 4x − [(4x)^{2}/2] + [(4x)^{3}/3] − [(4x)^{4}/4] +..........

= 4x − (16x^{2}/2) + (64x^{3}/3) − (256x^{4}/4) +............

= 4x − 8x^{2} + (64x^{3}/3) − 64x^{4} +............

Required condition is |x| < 1/4

**Question 2 :**

Write the first 4 terms of the logarithmic series

log(1 − 2x)

**Solution :**

log(1 − x) = −x − x^{2}/2 − x^{3}/3 − x^{4}/4 − ·· ·

Instead of x, apply 2x

log(1 - 2x) = -2x − [(2x)^{2}/2] - [(2x)^{3}/3] − [(2x)^{4}/4] +..........

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

= -2x − 2x^{2} + (8x^{3}/3) − 4x^{4} +............

Required condition is |x| < 1/2

**Question 3 :**

Write the first 4 terms of the logarithmic series

log [(1+3x)/(1−3x)]

**Solution :**

log [ (1 + x)/(1 − x)] = 2 [x + x^{3}/3 + x^{5}/5 +............]

Instead of x, apply 3x

log [ (1 + x)/(1 − x)]

= 2 [3x + (3x)^{3}/3 + (3x)^{5}/5 + (3x)^{7}/7............]

= 2 [3x + (27x^{3}/3) + (243x^{5}/5) + (2187x^{7}/7)+............]

Required condition is |x| < 1/3

**Question 4 :**

Write the first 4 terms of the logarithmic series

log [(1-2x)/(1+2x)]

**Solution :**

log [ (1 - x)/(1 + x)] = -2 [x + x^{3}/3 + x^{5}/5 +............]

Instead of x, apply 2x

log [(1-2x)/(1+2x)]

= -2 [2x + (2x)^{3}/3 + (2x)^{5}/5 + (2x)^{7}/7 +............]

= -2 [2x + (8x^{3}/3) + (32x^{5}/5) + (128x^{7}/7)+............]

Required condition is |x| < 1/2

**Question 5 :**

If y = x + x^{2}/2 + x^{3}/3 + x^{4}/4 + · · · , then show that x = y − y^{2}/2! + y^{3}/3! − y^{4}/4! + · · · .

**Solution :**

y = x + x^{2}/2 + x^{3}/3 + x^{4}/4 + · · ·

Let us take negative signs on both sides.

-y = -(x + x^{2}/2 + x^{3}/3 + x^{4}/4 + · · · )

-y = -x - x^{2}/2 - x^{3}/3 - x^{4}/4 - · · ·

-y = log (1 - x)

e^{-y} = 1 - x

x = 1 - e^{-y}

x = 1 - [1 - y/1! + y^{2}/2! - y^{3}/3! + ............]

x = 1 - 1 + y/1! - y^{2}/2! + y^{3}/3! - ............

x = y/1! - y^{2}/2! + y^{3}/3! - y^{4}/3 + ............

Hence proved.

**Question 6 :**

If p − q is small compared to either p or q, then show that

**Solution :**

Hence proved.

From this, we have to find the value of 8th root of (15/16)

p = 15, q = 16 and n = 8

= [15 (8 + 1) + 16 (8 - 1)]/[15 (8 - 1) + 16 (8 + 1)]

= [15 (9)+16(7)]/[15 (7) + 16 (9)]

= [135 + 112]/[105 + 144]

= 247/249

= 0.9919

**Question 7 :**

Find the coefficient of x^{4} in the expansion of (3−4x+x^{2})/e^{2x} .

**Solution :**

**e ^{x} = 1 + x/1! + x^{2}/2! + x^{3}/3! + ..............**

(3−4x+x^{2})/e^{2x }= (3−4x+x^{2})(1/e^{2x)}

Expansion for e^{x} :

**1 + x/1! + x ^{2}/2! + x^{3}/3! +**

Expansion for e^{2x} :

**1 - 2x/1! + (-2x) ^{2}/2! + (-2x)^{3}/3! +**

**1 - 2x/1! + 4x ^{2}/2 - 8x^{3}/6 +**

= (3−4x+x^{2}) (**1 - 2x + 2x ^{2} - 4x^{3}/3 +**

**Coefficient of x ^{4}**

= 2x^{4} + (16x^{4}/3) + 2x^{4}

= (2 + 16/3 + 2) x^{4}

= (4 + 16/3)x^{4}

= 28x^{4}/3

= (28/3)x^{4}

**Question 8 :**

Find the value

**Solution :**

Formula for

log [(1 + x)/(1 - x)] = 2 [x + x^{3}/3 + x^{5}/5 +..........]

[x + x^{3}/3 + x^{5}/5 +..........] = (1/2)log [(1 + x)/(1 - x)]

= (3/2) log(1 + (1/3)/(1-(1/3)) + (1/2)log(1 + (1/9)/(1-(1/9))

= (3/2) log(4/2) + (1/2)log(10/9)/(8/9)

= (3/2) log 2 + (1/2) log (5/4)

= (1/2) log 2^{3} + (1/2) log (5/4)

= (1/2) [log 8 + log (5/4)]

= (1/2) log 10

After having gone through the stuff given above, we hope that the students would have understood how to find expansion of logarithmic functions.

Apart from the stuff given above, if you need any other stuff in math, please use our google custom search here.

HTML Comment Box is loading comments...

You can also visit our following web pages on different stuff in math.

**WORD PROBLEMS**

**Word problems on simple equations **

**Word problems on linear equations **

**Word problems on quadratic equations**

**Area and perimeter word problems**

**Word problems on direct variation and inverse variation **

**Word problems on comparing rates**

**Converting customary units word problems **

**Converting metric units word problems**

**Word problems on simple interest**

**Word problems on compound interest**

**Word problems on types of angles **

**Complementary and supplementary angles word problems**

**Trigonometry word problems**

**Markup and markdown word problems **

**Word problems on mixed fractrions**

**One step equation word problems**

**Linear inequalities word problems**

**Ratio and proportion word problems**

**Word problems on sets and venn diagrams**

**Pythagorean theorem word problems**

**Percent of a number word problems**

**Word problems on constant speed**

**Word problems on average speed **

**Word problems on sum of the angles of a triangle is 180 degree**

**OTHER TOPICS **

**Time, speed and distance shortcuts**

**Ratio and proportion shortcuts**

**Domain and range of rational functions**

**Domain and range of rational functions with holes**

**Graphing rational functions with holes**

**Converting repeating decimals in to fractions**

**Decimal representation of rational numbers**

**Finding square root using long division**

**L.C.M method to solve time and work problems**

**Translating the word problems in to algebraic expressions**

**Remainder when 2 power 256 is divided by 17**

**Remainder when 17 power 23 is divided by 16**

**Sum of all three digit numbers divisible by 6**

**Sum of all three digit numbers divisible by 7**

**Sum of all three digit numbers divisible by 8**

**Sum of all three digit numbers formed using 1, 3, 4**

**Sum of all three four digit numbers formed with non zero digits**