Solve: ((2^x)-7)^2 = 1

The answer should be x = 2.58 (3 s.f.) or x = 3 but I have no idea how to get there. Thanks in advance!

1 answer

[ ( 2ˣ ) -7 )² = 1

Take the square root of both sides.

2ˣ - 7 = ±1

You must solve two equations:

2ˣ - 7 = -1 and 2ˣ - 7 = 1

1.)

2ˣ - 7 = -1

Add 7 to both sides

2ˣ - 7 + 7 = -1 + 7

2ˣ = 6

Take the logarithm of both sides.

x ∙ log ( 2 ) = log ( 6 )

x = log ( 6 ) / log ( 2 ) = 2.5849625

2.)

2ˣ - 7 = 1

Add 7 to both sides

2ˣ - 7 + 7 = 1 + 7

2ˣ = 8

Take the logarithm of both sides.

x ∙ log ( 2 ) = log ( 8 )

x = log ( 8 ) / log ( 2 ) = 3