solve for x:
1)log(x-3)+log(x+4)-logx=log5
To solve the given equation:
log(x-3) + log(x+4) - log(x) = log(5)
We can use the properties of logarithms to simplify the equation.
Step 1: Combine the logs using the properties of logarithms.
log[(x-3)(x+4)/x] = log(5)
Step 2: Apply the property that if two logarithms with the same base are equal, then their arguments are equal.
(x-3)(x+4)/x = 5
Step 3: Multiply both sides of the equation by x to eliminate the fraction.
(x-3)(x+4) = 5x
Step 4: Expand the equation and simplify.
x^2 + 4x - 3x - 12 = 5x
x^2 - 4x - 3x - 12 - 5x = 0
x^2 - 12x - 12 = 0
Step 5: Solve the quadratic equation.
To solve the quadratic equation, we can factor or use the quadratic formula.
Factoring:
(x - 6)(x + 2) = 0
Setting each factor equal to zero:
x - 6 = 0 OR x + 2 = 0
x = 6 OR x = -2
Therefore, the solutions to the equation are x = 6 and x = -2.