Solve the logarithmic equation for x. Log3(x^2 − 5x − 9) = 3
I Have no idea
Log3(x^2 − 5x − 9) = 3
By definition, 3^Log3(n) = n
Raise 3 to the power on both sides:
x^2 - 5x - 9 = 3^3 = 27
x^2 - 5x - 36 = 0
(x-9)(x+4) = 0
x = 9 or -4
Check:
9^2 - 5*9 - 9 = 81 - 45 - 9 = 27
(-4)^2 - 5*(-4) - 9 = 16 + 20 - 9 = 27
Log3(27) = 3
because 3^3 = 27
Don't worry! I will guide you step-by-step in solving the logarithmic equation.
Step 1: Move the logarithm to the other side of the equation.
log3(x^2 - 5x - 9) = 3
can be rewritten as
x^2 - 5x - 9 = 3^3
x^2 - 5x - 9 = 27
Step 2: Simplify the equation further.
x^2 - 5x - 9 = 27
Subtracting 27 from both sides gives
x^2 - 5x - 36 = 0
Step 3: Solve the quadratic equation for x.
This quadratic equation can be factored as
(x - 9)(x + 4) = 0
Setting each factor equal to zero gives two potential solutions:
x - 9 = 0 or x + 4 = 0
Solving each equation gives the values of x:
x = 9 or x = -4
So, the solutions to the logarithmic equation log3(x^2 - 5x - 9) = 3 are x = 9 and x = -4.
To solve the logarithmic equation Log3(x^2 − 5x − 9) = 3, you can follow these steps:
Step 1: Rewrite the equation in exponential form using the definition of logarithms. The logarithmic equation Log3(x^2 − 5x − 9) = 3 can be rewritten as 3^(Log3(x^2 − 5x − 9)) = 3^3.
Step 2: Simplify the left side of the equation using the property of logarithms. The base of the logarithm (3) and the base of the exponentiation (3) cancel out, leaving you with x^2 − 5x − 9 = 3^3 = 27.
Step 3: Rearrange the equation into a quadratic equation form by moving all terms to one side. x^2 − 5x − 9 - 27 = 0.
Step 4: Simplify the quadratic equation by combining like terms. x^2 - 5x - 36 = 0.
Step 5: Solve the quadratic equation by factoring, completing the square, or using the quadratic formula. In this case, the equation can be factored as (x + 4)(x - 9) = 0.
Step 6: Set each factor equal to 0 and solve for x. So, x + 4 = 0 or x - 9 = 0. This gives x = -4 or x = 9.
Therefore, the solutions to the logarithmic equation Log3(x^2 − 5x − 9) = 3 are x = -4 and x = 9.