Exponential Equations( 5) not easy

Example 1:

Solve each equation:

a)  (2) ^{x +1} +(2) ^{x - 1}   = 40                            

    

     (2) ^x ( 2¹ + 2^{- 1}) = 40

      

     (2) ^x  × ( 5/2) =40

     (2) ^x  =  40  × (2/5)

       (2) ^x   = 16

       (2) ^x  = 2⁴                  

        X = 4 

b) (3) ^{x +1} +(3) ^{x - 1}   = 30                           

    

     (3) ^x ( 3¹ + 3^{- 1}) = 30

      

     (3) ^x  × ( 10/3) =30

     (3) ^x  =  30  × (3/10)

       (3) ^x   = 9

       (3) ^x  = 3²                 

        X = 2

Example 2:

Solve each equation:

a)  (5) ^2x   - 26 (5) ^x    + 25   = 0                              

    

       ( 5^x -  1) (5^x -  25 )   = 0

      

      5^x =  1  or  5^x =  25

      X = 0     or   x = 2

b)  (3) ^2x   - 12 (3) ^x    + 27   = 0                              

    

       ( 3^x -  3) (3^x -  9 )   = 0

      

      3^x =  3  or  3^x =  9

      X = 1     or   x = 2

Example 3:

Solve each equation:

a)  (5)^{x- 1} =(x)^{x- 1}                                        

    

 x – 1 = 0       or   x = 5      or     x = - 5

    x = 1          or x = 5       or      x = - 5

b) (2)^{x- 3} =(x)^{x- 3}                                       

    

 x – 3 = 0       or    x = 2       

    x = 3          or    x = 2       

  

Example 4:

Solve each equation:

a)  (½) ^x +(½) ^x  + (½) ^x   =3/4                          

    

     (½) ^x  × ( 1 + 1 + 1 ) = 3/4

      (½) ^x   ×   3 = 3/4

       

      (½) ^x    = 3/4   ×  1/3

        (½) ^x    = 1/4

         (½) ^x    =  (½) ²   

          

                 X = 2 

b) (1/5) ^x +(1/5) ^x  + (1/5) ^x  = 3/125

       

 (1/5) ^x  ( …. + … + ….) =   ….

(1/5) ^x  ×   …..  = …..

(1/5) ^x   =   ……

 X = 3