(1/2)^x=32

(1/2)^x=32

Notice that (1/2) = 2^-1     and 32 = 2⁵ ......so we have.....

(2^-1)^x = 2⁵         so

2^-1x = 2⁵      now, just equate exponents

-1x = 5        divide by -1 on both sides

x = -5

(1/2) ^ x  =  32

First way:

If the unknown is an exponent, find the log of both sides:

log[ (1/2) ^ x ]  =  log( 32 )

x · log( 1/2 )  =  log( 32 )

x  =  log( 32 ) / log( 1/2 )

x  =  -5

Second way:

Write both sides with the same base:

Since  1/2  =  2^-1  and  32  =  2^5:

(2^-1)^x  =  2^5                             When you have an exponent to an exponent, multiply the exponents:

2^-x  =  2^5

-x  =  5

x  =  -5

To solve this, you must use logarithms.

x =  $${{log}}_{{\frac{{\mathtt{1}}}{{\mathtt{2}}}}}{\left({\mathtt{32}}\right)}$$

x = -5