Question

(1/2x)+1+log6 base5=log(5^(1/x)+125) find x​

Answer 1

Given :   (1/2x)  + 1  + log₅6 = log₅(5¹/ˣ + 125)

To find : x

Solution:

(1/2x)  + 1  + log₅6 = log₅(5¹/ˣ + 125)

=> (1/2x)  + 1   =  log₅(5¹/ˣ + 125) -  log₅6

=> (2x + 1) /2x =  log₅ ( (5¹/ˣ + 125)/6)

=>  (5¹/ˣ + 125)/6  = 5⁽²ˣ⁺¹⁾/²ˣ

=>  (5¹/ˣ + 125)  = 6 *  5⁽²ˣ⁺¹⁾/²ˣ

=> 5¹/ˣ + 5³ = 5*  5⁽²ˣ⁺¹⁾/²ˣ  +  5⁽²ˣ⁺¹⁾/²ˣ  

=> 5¹/ˣ + 5³ =    5⁽⁴ˣ⁺¹⁾/²ˣ  +  5⁽²ˣ⁺¹⁾/²ˣ  

Now there are two case

5¹/ˣ  =  5⁽⁴ˣ⁺¹⁾/²ˣ    & 5³ = 5⁽²ˣ⁺¹⁾/²ˣ  

or  5¹/ˣ   = 5⁽²ˣ⁺¹⁾/²ˣ   & 5³ =  5⁽⁴ˣ⁺¹⁾/²ˣ

Case 1 :

5¹/ˣ  =  5⁽⁴ˣ⁺¹⁾/²ˣ    & 5³ = 5⁽²ˣ⁺¹⁾/²ˣ  

=> 1/x = (4x + 1)/2x  => 4x + 1 = 2  => x =  1/4

 3 = (2x + 1)/2x  => 6x = 2x + 1 => x = x = 1/4

Hence x = 1/4

Case 2 :

5¹/ˣ   = 5⁽²ˣ⁺¹⁾/²ˣ   & 5³ =  5⁽⁴ˣ⁺¹⁾/²ˣ

=> 1/x = (2x + 1)/2x  => 2x + 1 = 2 => x = 1/2

  3 = (4x + 1)/2x  => 6x = 4x + 1  => x =  1/2

Hence x = 1/2

x = 1/4  & 1/2  are the Solution

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Answer 2

hope it helps...............