Mathematics LOGARITHMIC EQUATIONS

Logarithm-Definition :

Evety positive real number `N` can be expressed in exponential form as

`N=a^x` ..................(1)

e.g. `49=7^2`

where `'a'` is also a positive real different than unity and is called the base and `'x'` is called the exponent.

We can write the relation `(1)` in logatitlunic form as

`log_aN = x` ......... (2)

Hence the two relations

`a^x=N` and `log_a N=x`

are identical where `N>0,a>0, a ne 1`

Hence logarithm of a number to some base is the exponent by which the base must be raised in order to get that number. Logarithm of zero does not exist and logarithm of `(-)` ve reals are not defined in the system of real numbers.
i.e `a` is raised what power to get `N`

`text(Principal Properties of Logarithm :)`

If `m, n` are arbitrary positive real numbers where

`a>0 ; a ne 1`

`(1) \ \ \ \ \ \log_a m+log_an=log_a mn` `(m>0,n>0)`

`text(proof :)` Let `x_1=log_a m ; m=a^x`

`\ \ \ \ \ \x_2=log_a n ;n=a^(x_2)`

Now `\ \ \ \ \ \mn=a^(x_1).a^(x_2)`

`\ \ \ \ \ \mn=a^(x_1+x_2)`

`\ \ \ \ \ \x_1+x_2=log_amn`

(2) ` log_a (m/n)=log_a m-log_an`

`\ \ \ \ \ \m/n=a^(x_1-x_2)`

`\ \ \ \ \ \x_1-x_2=log_a (m/n)`

`\ \ \ \ \ \log_am-log_an=log_a (m/n)`

(3) `log_a m^x= x log_a m`

`\ \ \ \ \ \log_a m=p ; m=a^p`

`\ \ \ \ \ \m^x=a^(px)`

taking `log` both the side with base `a`

`\ \ \ \ \ \log_a m^x=log_a a^(px)=px=x log_a m`

Base Changing Theorem :

Can be stated as "quotient of the logarithm of two numbers is independent of their common base."

Symbolically, `(log_ca)/(log_cb)=log_ba`

proof: Let `log_c a=x; log_c b=y` & `log_ba=z`

`a=c^x; b=c^y, a=b^z`

`c^x=b^z`

`c^x=c^(yz)=> x=yz`

i.e., `z=x/y`

`log_b a=(log_c a)/(log_c b)`

Case-I :

`log_b a=1/(log_a b)`

We have proved that `(log_c a)/(log_c b)=log_b a`

put `c=a`

Similarly `(log_a a)/(log_a b)=log_b a`

or `1/(log_ab)=log_b a`

Case-II :

`(log_ba) . (log_c b) . (log_d c)=log_d a`

proof `(log a)/(log b) xx(log b)/(log c) . (log c)/(log d)=(log a)/(log b)=log_d a`

Case-III : Very imp form :

`a^(log_b c)=c^(log_b a)`

proof `a^(log_b c)=a^((log_b c)(log_c a))=a^((log_a c)(log_b c).(log_c a))=c^(log_b a)`

`=> a^(log_b c)=c^(log_b a)`

Exponential Equations

If we have an equation of the form `a^x= b(a> 0),` then

`(i) x in phi `,if `b <= 0`
`(ii) x = log_a b,` if `b > 0, a ne 1`
`(iii) x in phi,` if `a = 1, b ne 1`
`(iv) x in R,` if `a= 1, b = 1` (since, `1^x = 1 => 1 = 1, x in R)`

`text(Some Standard Forms to Solve Exponential Equations)`

`text(Form 1)` An equation in the form
`a^(f(x)) = 1, a > 0, a ne 1` is equivalent to the equation `f(x) = 0`

`text(Form 2)` An equation in the form
`f(a^x) = 0`
is equivalent to the equation `f(t) = 0`, where `t =a^x.`
If `t_1, t_2, t_3, ... , t_k` are the roots of `f( t) = 0,` then
`a^k = t_1 a^x = t_2 a^x = t_3 ... a^x = t_k`

`text(Form 3)` An equation of the form
`alphaa^(f(x)) + betab^(f(x)) + yc^(f(x)) = 0,`
where `alpha,beta , gamma ne 0` and the bases satisfy the condition `b^2 = ac` is
equivalent to the equation
`at^2 + betat + y = 0,` where `t =(a // b) f(x)`
If roots of this equation are `t_1` and `t_2` , then
`(a // b)f(x) = t_1` and `(a // b)f(x) = t_2`

`text(Form 4)` An equation in the form
`alpha. a^(f(x)) + beta. b^(f(x)) + c = 0,`

where `alpha,beta, c in R` and `ab = 1` (a and b are inverse +ve numbers)
is equivalent to the equation
`at^2 + ct + beta = 0,` where `t = a^(f(x))`.
If roots of this equation are `t_1` and `t_2` then `a^(f(x)) = t_1` and `a^(f(x)) = t_2 .`

`text(Form 5 ) ` An equation of the form
`a^(f(x)) + b^(f(x)) = c,`
where `a, b, c in R` and `a, b, c` satisfies the condition `a^2 + b^2 = c,` then solution
of this equation is `f(x) = 2` and no other solution of this equation.

`text(Form 6)` An equation of the form `{f(x)}^(g (x))` is equivalent to the equation
`{f(x)}^(g(x)) = 10^(g(x)Iogf(x)),` where `f(x) > 0.`

Absolute value function :

(a) `y=|x|= x` if `x ge 0` and `-x` if `x <0`

(b) `sqrt (x^2) =|x|`

(c) `log x^(2n)=2n log|x|`, where `n in I`

Equations Containing Absolute Values:

By definition, `| x | = x,` if `x>= 0`

`| x | = - x,` if `x<= 0`

Important Forms Containing Absolute Values :

Form 1 The equation of the form `| f(x) + g(x) | = | f(x) | +| g(x) |`

is equivalent of the system `f(x)g(x)>= 0.`

lnequations Containing Absolute Values:

`|x| < a => -a < x < a` (a>0)

`|x| =< a => - a <= x <= a`

`|x| > a => x < - a` and `x > -a`

`|x| >= a => x <= -a` and `x >= -a`