Some important Logarithmic and Exponential formulae

1. $\quad$ If $\mathrm{a}^{\mathrm{x}}=\mathrm{y}$, then $\mathrm{x}=\log _{\mathrm{a}} \mathrm{y}$

2. $\quad \log _{\mathrm{a}} \mathrm{a}=1 \& \log _{\mathrm{a}} 1=0$

3. $\quad \mathrm{a}^{\log _{\mathrm{a}} \mathrm{n}}=\mathrm{n}$

4. $\quad \log _{\mathrm{a}} \mathrm{mn}=\log _{\mathrm{a}} \mathrm{m}+\log _{\mathrm{a}} \mathrm{n}$

5. $\quad \log _{a}\left(\frac{m}{n}\right)=\log _{a} m-\log _{a} n$

6 $\quad \log _{\mathrm{a}} \mathrm{m}^{\mathrm{n}}=\mathrm{n} \log _{\mathrm{a}} \mathrm{m}$

7. $\quad \log _{\mathrm{b}} \mathrm{a}=\frac{\log _{\mathrm{c}} \mathrm{a}}{\log _{\mathrm{c}} \mathrm{b}}$

8. $\quad \log _{a^{n}} a^{m}=\frac{m}{n}$

9. $\quad \log _{\mathrm{b}} \mathrm{a}=\frac{1}{\log _{\mathrm{a}} \mathrm{b}}$

10. $\quad \log _{e}(1+x)=x-\frac{x^{2}}{2}+\frac{x^{3}}{3}-\frac{x^{4}}{4}+\ldots \ldots \ldots \ldots$

11. $\quad \log _{e}(1-x)=-x-\frac{x^{2}}{2}-\frac{x^{3}}{3}-\frac{x^{4}}{4}$

12. $\quad \log _{e}(1+x)(1-x)=\log _{e}\left(1-x^{2}\right)=-2\left(\frac{x^{2}}{2}+\frac{x^{4}}{4}+\frac{x^{6}}{6}+\ldots \ldots \ldots \ldots . ..\right)$

13. $\quad \log _{e}\left(\frac{1+x}{1-x}\right)=2\left(x+\frac{x^{3}}{3}+\frac{x^{5}}{5}+\ldots \ldots \ldots \ldots \ldots \ldots . . . .\right.$.

14. $\quad \log _{\mathrm{e}} 2=1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}……..$

15. $\quad \mathrm{e}^{\mathrm{x}}=1+\frac{\mathrm{x}}{1 !}+\frac{\mathrm{x}^{2}}{2 !}+\frac{\mathrm{x}^{3}}{3 !}+………$

16. $\quad \mathrm{e}^{\mathrm{x}}+\mathrm{e}^{-\mathrm{x}}=2\left(1+\frac{\mathrm{x}^{2}}{2 !}+\frac{\mathrm{x}^{4}}{4 !}+\ldots \ldots \ldots \ldots\right)$

17. $\quad e^{x}-e^{-x}=2\left(\frac{x}{1 !}+\frac{x^{3}}{3 !}+\frac{x^{5}}{5 !} \ldots \ldots \ldots \ldots . ..\right)$

18. $\quad \mathrm{e}=1+\frac{1}{1 !}+\frac{1}{2 !}+\frac{1}{3 !}+………$

$\quad\mathrm{e}$ is an irrational number $\&$ it lies between $2 \& 3 . \mathrm{e} \cong 2.7183$

19. $\quad a^{y}=e^{y \log _{e} a}=1+y\left(\log _{e} a\right)+\frac{y^{2}}{2 !}\left(\log _{e} a\right)^{2}+\frac{y^{3}}{3 !}\left(\log _{e} a\right)^{3}+………$

20. $\quad \log _{2} \log _{2} \sqrt{\sqrt{\sqrt{\sqrt{\ldots \ldots \ldots \ldots \ldots . . . \sqrt{2}}}}}=\mathrm{n}$ where $\mathrm{n}$ is the number of square roots

21. $\quad \mathrm{a}^{\sqrt{\log _{\mathrm{a}} \mathrm{b}}}=\mathrm{b}^{\sqrt{\log _{\mathrm{b}} \mathrm{a}}}$

Vn-Method

(i) To find the sum of series of the form

$\frac{1}{\mathrm{a} _{1} \mathrm{a} _{2} \ldots \ldots \ldots \ldots \mathrm{a} _{\mathrm{r}}}+\frac{1}{\mathrm{a} _{2} \mathrm{a} _{3} \ldots \ldots \ldots \ldots \mathrm{a} _{\mathrm{r}+1}}+\ldots \ldots \ldots \ldots \ldots \ldots+\frac{1}{\mathrm{a} _{\mathrm{n}} \mathrm{a} _{\mathrm{n}+1} \ldots \ldots \ldots \ldots \mathrm{a} _{\mathrm{n}+\mathrm{r}-1}}$ where $\mathrm{a} _{1}, \mathrm{a} _{2}$,

$\left( \text { Here }T_{n}=\frac{1}{a _{n} a _{n+1} \ldots \ldots \ldots \ldots a _{n+r-1}}\right)$

……..are in A.P. Let $V _{n}=\frac{1}{a _{n+1} a _{n+2} \ldots \ldots \ldots . a _{n+r-1}}$ (avoiding first term for $V _{n}$ ie $a _{n}$ in $T _{n}$ )

$\mathrm{T} _{\mathrm{n}}=\frac{-1}{\mathrm{~d}(\mathrm{r}-1)}\left(\mathrm{V} _{\mathrm{n}}-\mathrm{V} _{\mathrm{n}-1}\right)=\frac{1}{\mathrm{~d}(\mathrm{r}-1)}\left(\mathrm{V} _{\mathrm{n}-1}-\mathrm{V} _{\mathrm{n}}\right)$

put $\mathrm{n}=1,2,3 \ldots \ldots \ldots \ldots . . \mathrm{n}$ and add to get $\mathrm{S} _{\mathrm{n}}$.

$\mathrm{S} _{\mathrm{n}}=\mathrm{T} _{1}+\mathrm{T} _{2}+\ldots \ldots . \mathrm{T} _{\mathrm{n}}=\frac{1}{\mathrm{~d}(\mathrm{r}-1)}\left(\mathrm{V} _{0}-\mathrm{V} _{\mathrm{n}}\right)$

$ =\frac{1}{d(r-1)}\left(\frac{1}{a _{1} a _{2} \ldots \ldots . a _{r-1}}-\frac{1}{a _{n+1} a _{n+2} \ldots \ldots a _{n+r-1}}\right) $

Eg. (a) $\quad \frac{1}{1.2 .3}+\frac{1}{2.3 .4}+\ldots \ldots \ldots \ldots \ldots+\frac{1}{n(n+1)(n+2)}=\frac{1}{1 .(3-1)}\left(\frac{1}{1.2}-\frac{1}{(n+1)(n+2)}\right)$

$=\frac{1}{4}-\frac{1}{2(\mathrm{n}+1)(\mathrm{n}+2)}$

(b) $\quad \frac{1}{1 \cdot 2 \cdot 3.4}+\frac{1}{2 \cdot 3 \cdot 4.5}+\ldots \ldots \ldots . \frac{1}{\mathrm{n}(\mathrm{n}+1)(\mathrm{n}+2)(\mathrm{n}+3)}$

$ =\frac{1}{1 .(4-1)}\left(\frac{1}{1.2 \cdot 3}-\frac{1}{(n+1)(n+2)(n+3)}\right) $

(ii) Summation of series of the form $\mathrm{a} _{1} \mathrm{a} _{2} \ldots \ldots \mathrm{a} _{\mathrm{r}}+\mathrm{a} _{2} \mathrm{a} _{3} \ldots \ldots . \mathrm{a} _{\mathrm{r}+1}+$. $\mathrm{a} _{1}, \mathrm{a} _{2} \ldots . .$. are in A.P

Here $T _{n}=a _{n} a _{n+1} \cdots \cdots \ldots . a _{n+r-1}$

Let $V _{n}{ } _{n}=a _{n} a _{n+1} \cdots \ldots \ldots \ldots \ldots a _{n-r+1} a _{n-r}\left(\right.$ Take one term extra in $T _{n}$ for $\left.V _{n}\right)$

$\mathrm{T} _{\mathrm{n}}=\frac{1}{(\mathrm{r}+1) \mathrm{d}}\left(\mathrm{V} _{\mathrm{n}}-\mathrm{V} _{\mathrm{n}-1}\right)$.

Put $\mathrm{n}=1,2,3 \ldots \ldots .$. and add to get $\mathrm{S} _{\mathrm{n}}$

$S _{n}=T _{1}+T _{2}+\ldots \ldots \ldots \ldots+T _{n}=\frac{1}{d(r+1)}\left(V _{n}-V _{0}\right)=\frac{1}{d(r+1)}\left(a _{n} a _{n+1} \ldots \ldots \ldots . a _{n+r}-a _{0} a _{1} a _{2} \ldots \ldots \ldots \ldots a _{r}\right)$ where $\mathrm{a} _{0}=\mathrm{a} _{1}-\mathrm{d}$

Eg (a) $\quad 1.2+2.3+\ldots \ldots \ldots \ldots \ldots \ldots . .+\mathrm{n}(\mathrm{n}+1)=\frac{1}{1 .(2+1)}((\mathrm{n}+1)(\mathrm{n}+2)-0.1 .2)=\frac{\mathrm{n}(\mathrm{n}+1)(\mathrm{n}+2)}{3}$

(b) $\quad 1.2 .3 .4+2.3 .4 .5+$ $+\mathrm{n}(\mathrm{n}+1)(\mathrm{n}+2)(\mathrm{n}+3)$

$ \begin{aligned} & =\frac{1}{1 .(4+1)}(\mathrm{n}(\mathrm{n}+1)(\mathrm{n}+2)(\mathrm{n}+3)(\mathrm{n}+4)-0 \cdot 1 \cdot 2 \cdot 3 \cdot 4) \\ & =\frac{1}{5} \mathrm{n}(\mathrm{n}+1)(\mathrm{n}+2)(\mathrm{n}+3)(\mathrm{n}+4) \end{aligned} $

Solved examples

1. If the sides of a triangle are in A.P and the greatest angle of the triangle is double the smallest, then the ratio of sides of the triangle is

(a) $3: 4: 5$

(b) $4: 5: 6$

(c) $5: 6: 7$

(d) None of these

Show Answer

Solution :

Applying sine rule, we have

$\frac{\mathrm{a}-\mathrm{d}}{\sin \theta}=\frac{\mathrm{a}}{\sin (\pi-3 \theta)}=\frac{\mathrm{a}+\mathrm{d}}{\sin 2 \theta}$

$\Rightarrow \frac{\mathrm{a}-\mathrm{d}}{\sin \theta}=\frac{\mathrm{a}}{3 \sin \theta-4 \sin ^{3} \theta}=\frac{\mathrm{a}+\mathrm{d}}{2 \sin \theta \cos \theta}$

$\Rightarrow \frac{\mathrm{a}-\mathrm{d}}{1}=\frac{\mathrm{a}}{3-4 \sin ^{2} \theta}=\frac{\mathrm{a}+\mathrm{d}}{2 \cos \theta}$ gives $\cos \theta=\frac{\mathrm{a}+\mathrm{d}}{2(\mathrm{a}-\mathrm{d})}$

Also, $3-4 \sin ^{2} \theta=\frac{a}{a-d}$

$\Rightarrow 3-4\left(1-\cos ^{2} \theta\right)=\frac{a}{a-d}$ gives $-1+\left(\frac{a+d}{a-d}\right)^{2}=\frac{a}{a-d}$

$\Rightarrow \frac{4 \mathrm{ad}}{\mathrm{a}-\mathrm{d}}=\mathrm{a} \Rightarrow 4 \mathrm{ad}=\mathrm{a}^{2}-\mathrm{ad}$ gives $\mathrm{a}=5 \mathrm{~d}$

$\therefore$ sides $\mathrm{a}-\mathrm{d}: \mathrm{a}: \mathrm{a}+\mathrm{d}$

$5 \mathrm{~d}-\mathrm{d}: 5 \mathrm{~d}: 5 \mathrm{~d}+\mathrm{d}=4: 5: 6$

Answer: (b)

2. The sum of

$\frac{3}{1.2} \cdot \frac{1}{2}+\frac{4}{2.3}\left(\frac{1}{2}\right)^{2}+\frac{5}{3.4} \cdot\left(\frac{1}{2}\right)^{3}+\ldots \ldots \ldots \ldots . . \mathrm{n}$ terms is

(a) $1-\frac{1}{(\mathrm{n}+1) 2^{\mathrm{n}}}$

(b) $1-\frac{1}{\mathrm{n} \cdot 2^{\mathrm{n}-1}}$

(c) $1+\frac{1}{(\mathrm{n}+1) 2^{\mathrm{n}}}$

(d) None of these

3. Coefficient of $x^{49}$ in the expansion of $(x-1)(x-3)(x-5)……….(x-99)$ is

(a) $-99^{2}$

(b) $1$

(c) $-2500$

(d) None of these

4. The coefficients of $x^{15}$ in the product

$(1-\mathrm{x})(1-2 \mathrm{x})\left(1-2^{2} \mathrm{x}\right) ………\left(1-2^{15} x\right)$ is

(a) $2^{105}-2^{121}$

(b) $2^{121}-2^{105}$

(c) $2^{120}-2^{104}$

(d) None of these

5. The sum to $2 \mathrm{n}$ terms of the series

$\frac{3}{4}+\frac{7}{4}+\frac{15}{16}+\frac{31}{16}+\frac{63}{64}+\frac{127}{64}+………..$ is

(a) $3 \mathrm{n}-\frac{2}{3}\left(1-\frac{1}{4^{\mathrm{n}}}\right)$

(b) $3 \mathrm{n}-\frac{10}{21}\left(1-\frac{1}{4^{\mathrm{n}}}\right)$

(c) $3 n-\frac{13}{21} \frac{1}{4^{n}}$

(d) None of these

6. The sum to $\mathrm{n}$ terms of the series

$\frac{1}{1-\frac{1}{4}}+\frac{1}{(1+3)-\frac{1}{4}}+\frac{1}{(1+3+5)-\frac{1}{4}}+\ldots \ldots \ldots \infty$

(a) $\frac{2 \mathrm{n}}{2 \mathrm{n}+1}$

(b) $\frac{4 n}{2 n+1}$

(c) $\frac{2}{2 \mathrm{n}+1}$

(d) None

7. The sum of the series

$\frac{1}{1.3}+\frac{2}{1.3 .5}+\frac{3}{1.3 .5 .7}+………$ $\infty$ is

(a) $1$

(b) $\frac{1}{2}$

(c) $\frac{3}{2}$

(d) None

Practice questions

1. Let $S _{1}, S _{2}, \ldots \ldots \ldots \ldots . . .$ be squares such that for each $n \geq 1$ the length of a side of $S _{n}$ equals the length of a diagonal of $\mathrm{S} _{\mathrm{n}+1}$. If the length of a side of $\mathrm{S} _{1}$ is $10 \mathrm{~cm}$, then for which of the following values of $\mathrm{n}$ is the area of $\mathrm{S} _{\mathrm{n}}$ less than $1 \mathrm{sq} . \mathrm{cm}$.

(a) 7

(b) 8

(c) 9

(d) 10

2. If $\mathrm{a}, \mathrm{b}, \mathrm{c}$ are is A.P and $\mathrm{a}^{2}, \mathrm{~b}^{2}, \mathrm{c}^{2}$ are in H.P, then $\mathrm{b}^{2}=$

(a) $\frac{\mathrm{ca}}{2}$

(b) $2 \mathrm{ca}$

(c) $\frac{-\mathrm{ca}}{2}$

(d) $-2 \mathrm{ca}$.

3. Let the $\mathrm{HM} \& \mathrm{GM}$ of two positive numbers $\mathrm{a} \& \mathrm{~b}$ be in the ratio $4: 5$ then $\mathrm{a}: \mathrm{b}$ is

(a) $1: 2$

(b) $2: 3$

(c) $3: 4$

(d) $1: 4$

4. If $\cos (x-y), \cos x, \cos (x+y)$ are in H.P., then the value of $\cos x \sec \frac{y}{2}$ is

(a) $\pm 1$

(b) $\pm \frac{1}{\sqrt{2}}$

(c) $\pm \sqrt{2}$

(d) $\pm \sqrt{3}$

5. If $x \& y$ are positive real numbers and $m, n$ are positive integers, then the minimum value of $\frac{x^{m} y^{n}}{\left(1+x^{2 m}\right)\left(1+y^{2 n}\right)}$ is

(a) $2$

(b) $\frac{1}{4}$

(c) $\frac{1}{2}$

(d) $1$

6. There are two numbers $\mathrm{a} \& \mathrm{~b}$ whose product is 192 and the quotient of A.M. by H.M. of their greatest common divisor and least common multiple is $\frac{169}{48}$. The smaller of $\mathrm{a} \& \mathrm{~b}$ is

(a) 2

(b) 4

(c) 6

(d) 12

7. Consider the sequence $1,2,2,3,3,3,………$ where n occurs $n$ times. The number that occurs as $2007^{\text {th }}$ term is

(a) 61

(b) 62

(c) 63

(d) 64

8. Read the following paragraph and answer the questions.

Let $A _{1}, G _{1}, H _{1}$ denote the A.M., G.M., H.M of two distinct positive numbers.

For $n \geq 2$, let $A _{n-1}$ and $H _{n-1}$ have A.M., G.M., H.M as $A _{n}, G _{n}, H _{n}$ respectively.

(i) Which of the following statements is correct?

(a) $\mathrm{G} _{1}>\mathrm{G} _{2}>\mathrm{G} _{3}>………….$

(b) $\mathrm{G} _{1}<\mathrm{G} _{2}<\mathrm{G} _{3}<………….$

(c) $\mathrm{G} _{1}=\mathrm{G} _{2}=\mathrm{G} _{3}=………….$.

(d) $\mathrm{G} _{1}<\mathrm{G} _{3}<\mathrm{G} _{5}………….$. and $\mathrm{G} _{2}>\mathrm{G} _{4}>\mathrm{G} _{6}>………….$.

(ii) Which are of the following statement is correct?.

(a) $\mathrm{A} _{1}>\mathrm{A} _{2}>\mathrm{A} _{3}>………….$.

(b) $\mathrm{A} _{1}<\mathrm{A} _{2}<\mathrm{A} _{3}<………….$

(c) $\mathrm{A} _{1}>\mathrm{A} _{3}>\mathrm{A} _{5}>………….$ and $\mathrm{A} _{2}<\mathrm{A} _{4}<\mathrm{A} _{6}<………..$

(d) $\mathrm{A} _{1}<\mathrm{A} _{3}<\mathrm{A} _{5}<………….$ and $\mathrm{A} _{2}>\mathrm{A} _{4}>\mathrm{A} _{6}>………..$

(iii) Which are of the following statement is correct?

(a) $\mathrm{H} _{1}>\mathrm{H} _{2}>\mathrm{H} _{3}>………….$

(b) $\mathrm{H} _{1}<\mathrm{H} _{2}<\mathrm{H} _{3}<………….$

(c) $\mathrm{H} _{1}>\mathrm{H} _{3}>\mathrm{H} _{5}>………….$ and $\mathrm{H} _{2}<\mathrm{H} _{4}<\mathrm{H} _{6}<………..$

(d) $\mathrm{H} _{1}<\mathrm{H} _{3}<\mathrm{H} _{5}<………….$ and $\mathrm{H} _{2}>\mathrm{H} _{4}>\mathrm{H} _{6}>………..$

9. If $x, y, z>0$ and $x+y+z=1$, the $\frac{x y z}{(1-x)(1-y)(1-z)}$ is necessarily

(a) $\geq 8$

(b) $\leq \frac{1}{8}$

(c) $<\frac{1}{8}$

(d) None of these

10. Match the following:

11. The coefficient of $\mathrm{x}^{203}$ is the expansion of $(\mathrm{x}-1)\left(\mathrm{x}^{2}-2\right)\left(\mathrm{x}^{3}-3\right)$ … $\left(\mathrm{x}^{20}-20\right)$ is

(a) $-35$

(b) $21$

(c) $13$

(d) $25$

12. Read the following passage and answer the questions :-

Let $A B C D$ be a unit square and $0<\alpha<1$. Each side of the square is divided in the ratio $\alpha: 1-\alpha$, as shown in the figure. These points are connected to obtain another square. The sides of new square are divided in the ratio $\alpha: 1-\alpha$ and points are joined to obtain another square. The process is continued indefinitely.

Let $a _{n}$ denote the length of side and $A _{n}$ the area of the nth square

(i) The value of $\alpha$ for which $\sum _{n=1}^{\infty} A _{n}=\frac{8}{3}$ is

(a) $1 / 3,2 / 3$

(b) $1 / 4,3 / 4$

(c) $1 / 5,4 / 5$

(d) $1 / 2$

(ii) The value of $\alpha$ for which side of $n^{\text {th }}$ square equals the diagonals of $(n+1)$ th square is

(a) $1 / 3$

(b) $1 / 4$

(c) $1 / 2$

(d) $1 / \sqrt{2}$

(iii) If $a=1 / 4$ and $P _{n}$ denotes the perimeter of the $n^{\text {th }}$ square then $\sum _{n=1}^{\infty} P _{n}$ equals

(a) $8 / 3$

(b) $32 / 3$

(c) $16 / 3$

(d) $\frac{8}{3}(4+\sqrt{10})$