(1/4)^x+2=16 x=?

(1/4)^x+2=16 x=? someone please help. i need to know how this is done, step by step.

 

x≈0.174098.

P.S. There is no direct solution for this type of problems. Instead, the computers use iterations and interpolations to find the value of x.

The solution depend upon whether you meant (1/4)^x + 2  =  16  or  (1/4)^{x  + 2}  =  16 ...

 

      (1/4)^x + 2  =  16     

            (1/4)^x  =  14                      Since the unknown is an exponent, find the log of both sides:

   log( (1/4)^x )  =  log(14)               A property of logs allows the exponent to come out as a multiple:

    x · log(1/4)  =  log(14)

                    x  =  log(14) / log(1/4)     --->     x = -1.9036...

 

(1/4)^{x +  2}  =  16                             Since 1/4  =  4^-1:

 (4^-1)^{x + 2}  =  16                             Since you have an exponent to an exponent, multiply the exponents:

      4^{-x - 2}  =  4²                              Since the bases are equal, the exponents must be equal:

       -x - 2  =  2

            -x  =  4

             x  =  -4