For what values of x is the following statement true?
log42x2 = 1 – log422x – 1
- -2, 4
- 2, -4
- -1, -3
- 1, 3
- 1,-3
log42x2 = 1 – log422x – 1
→ log42x2 + log422x – 1 = 1
→ log4(2x2 · 22x – 1) = 1
→ 41 = 2x2 ·22x – 1
→ 2x2 + 2x – 1 = 41 = 22
→ x2 + 2x – 1 = 2
→ x2 + 2x – 3 = 0
→ (x + 3)(x – 1) = 0
→ x = 1 or x = -3
→ log4(2x2 · 22x – 1) = 1
→ 41 = 2x2 ·22x – 1
→ 2x2 + 2x – 1 = 41 = 22
→ x2 + 2x – 1 = 2
→ x2 + 2x – 3 = 0
→ (x + 3)(x – 1) = 0
→ x = 1 or x = -3
The answer is (e).
Another way to go, which is a bit more, um, cerebral — a bit more, uh, earth-bound, is to take each side separately.
Set log42x2 = k and get an expression for k. Then set 1 – log422x – 1 = k (the same k), and get another expression for k. Then set the two expressions for k equal to each other and go from there.
You can also note that 1 = log44 which puts the right-hand expression on an interesting path.