Question

Find the value of x that satisfies the equation log_2(x²-9)=3log_2 2+1

Answer 1

Starting with the equation:

log₂(x² - 9) = 3log₂(2) + 1

We can simplify the right side by using the property of logarithms that states loga(b^c) = cloga(b).
This allows us to rewrite 3log₂(2) as log₂(2³), which equals log₂(8).

Our equation now becomes:

log₂(x² - 9) = log₂(8) + 1

Using the property that loga(b) = loga(c) if and only if b = c, we can eliminate the logarithms and set the expressions inside equal to each other:

x² - 9 = 8 + 1

Combining like terms:

x² - 9 = 9

Adding 9 to both sides:

x² = 18

Taking the square root of both sides:

x = ±√18

Simplifying the square root of 18:

x = ±√(9 * 2)

x = ±3√2

Therefore, the values of x that satisfy the equation are x = 3√2 and x = -3√2.