To solve this equation, first distribute the -4 to 2x² – 14:
log2x – 8x² + 56 = 2
Next, bring all the terms to one side:
log2x – 8x² + 54 = 0
Now, we use a logarithmic identity:
log(base a) x = y is equivalent to x = a^y
Using this identity with base 2 and y = the entire expression, we can rewrite the equation as:
2x – 8x² + 54 = 0
At this point, we have a quadratic equation. We can solve for x by using either factoring, completing the square, or the quadratic formula. Since the quadratic in this case is quadratic being in standard form we will solve it using the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / 2a
In this case, a = -8, b = 2, and c = 54:
x = (-2 ± √(2^2 - 4(-8)(54))) / 2(-8)
x ≈ 2.162 and x ≈ -1.286
We plug each of these x-values back into the original equation to check for extraneous solutions. In this case, only x ≈ 2.162 works. Therefore, the solution to the equation is:
x ≈ 2