Solve for x: log3(x^2-4)-log3(x+2)=2
log3(x^2-4)-log3(x+2)=2
log3 [(x^2-4)/x+2)] = 2
log3 [(x+2)(x-2)/(x+2)] = 2
log3 (x-2) = 2
x-2 = 3^2
x = 11
log3 (x^2-4) -log3 (x+2)=2
log3 [(x^2-4)/x+2)] = 2
log3 [(x+2)(x-2)/(x+2)] = 2
log3 (x-2) = 2
x-2 = 3^2
x = 11
Adam, why would you just cut-and-paste my solution and claim it as yours ?
You didn't even bother to fix my error in the 2nd line, where I missed a bracket.
To solve the equation log3(x^2-4) - log3(x+2) = 2, we can use logarithmic properties to simplify the equation before solving for x.
Step 1: Combine the logarithms using the quotient rule. According to the quotient rule for logarithms, loga(b) - loga(c) = loga(b/c). Applying this rule to our equation, we get:
log3((x^2-4)/(x+2)) = 2
Step 2: Convert the logarithmic equation into exponential form. The equation loga(b) = c can be rewritten as a^c = b. Applying this conversion to our equation, we have:
3^2 = (x^2-4)/(x+2)
Step 3: Simplify the equation. Evaluate 3^2 to get 9:
9 = (x^2-4)/(x+2)
Step 4: Multiply both sides of the equation by x+2 to eliminate the fraction:
9(x+2) = x^2 - 4
Step 5: Expand and rearrange the equation:
9x + 18 = x^2 - 4
Step 6: Move all terms to one side to create a quadratic equation:
0 = x^2 - 9x - 4 - 18
0 = x^2 - 9x - 22
Step 7: Solve the quadratic equation. This equation can be factored or solved using the quadratic formula. In this case, let's use the quadratic formula:
x = (-b ± √(b^2 - 4ac))/(2a)
For our equation x^2 - 9x - 22 = 0, the coefficients are:
a = 1
b = -9
c = -22
Plugging these values into the quadratic formula, we have:
x = (-(-9) ± √((-9)^2 - 4 * 1 * (-22))) / (2 * 1)
Simplifying further:
x = (9 ± √(81 + 88)) / 2
x = (9 ± √169) / 2
x = (9 ± 13) / 2
The two possible solutions are:
x1 = (9 + 13) / 2 = 22/2 = 11
x2 = (9 - 13) / 2 = -4/2 = -2
So, the values of x that satisfy the equation log3(x^2-4) - log3(x+2) = 2 are x = 11 and x = -2.