solve for x

Question

2^x=0.125

5^2x=39

13^x=27

Answer 1

2 ^ x = 0.125

Take the logarithm to the base 2 of both sides:

log ( a ^ n ) = n * log ( a )

log( base 2 ) [ 2 ^ x ] = log( base 2 ) ( 2 ) * x = 1 * x = x

Becouse: log( base 2 ) ( 2 )

0.125 = 1 / 8 = 1 / 2 ^ 3 = 2 ^ ( - 3 )

log( base 2 ) [ 2 ^ ( - 3 ) ] = - 3

log( base 2 ) [ 2 ^ x ] = - 3

5 ^ ( 2 x ) = 39

2 x * log ( 5 ) = log ( 39 ) Divide both sides by ] 2 * log ( 5 ) ]

x = log ( 39 ) / [ 2 log ( 5 ) ]

x = 1.13815

13 ^ x = 27

x * log ( 13 ) = log ( 27 ) Divide both sides by log ( 13 )

x = log ( 27 ) / log ( 13 )

x = 1.28495

Answer 2

log( base 2 ) [ 2 ^ ( - 3 ) ] = - 3

x = -3