Solve Log(x^2 + 4) = 2 + log x -log 20
log(x^2 + 4) = 2 + log x - log 20
x^2+4 = 100 * x / 20
x^2 + 4 = 5x
x^2-5x+4 = 0
(x-1)(x-4) = 0
x = 1 or 4
The solutions to the equation Log(x^2 + 4) = 2 + log x - log 20 are x = 1 or x = 4.
To solve the equation log(x^2 + 4) = 2 + log x - log 20, we need to simplify the equation and isolate the variable x.
Step 1: Combine the logarithms on the right side of the equation using logarithmic properties.
log(x^2 + 4) = log x^1 - log 20
Step 2: Apply the exponent property of logarithms to the x terms on the right side of the equation.
log(x^2 + 4) = log (x/20)
Step 3: Set the arguments of the logarithms equal to each other since the logarithms have the same base.
x^2 + 4 = x/20
Step 4: Multiply both sides of the equation by 20 to eliminate the fraction.
20(x^2 + 4) = x
Step 5: Distribute 20 through the parenthesis on the left side of the equation.
20x^2 + 80 = x
Step 6: Rearrange the equation into quadratic form by moving all terms to one side.
20x^2 - x + 80 = 0
Step 7: Solve the quadratic equation by factoring, completing the square, or using the quadratic formula. Since this equation does not factor easily, we will use the quadratic formula.
x = (-b ± √(b^2 - 4ac)) / (2a)
In this case, a = 20, b = -1, and c = 80.
x = (-(-1) ± √((-1)^2 - 4(20)(80))) / (2(20))
x = (1 ± √(1 - 3200)) / 40
Step 8: Simplify the square root expression inside the formula.
x = (1 ± √(-3199)) / 40
As the square root of a negative number is imaginary, this equation does not have real solutions. Therefore, there is no real value for x that satisfies the equation.
We can start by using the properties of logarithms to simplify the equation.
First, we can combine the logs on the right side:
log(x^2 + 4) = log(x) + log(20/1)
log(x^2 + 4) = log(20x)
Next, we can use the fact that log b + log c = log (b * c):
log(x^2 + 4) = log(20x)
By setting the expressions inside the logs equal to each other, we get:
x^2 + 4 = 20x
Rearranging the equation:
x^2 - 20x + 4 = 0
Now we have a quadratic equation. We can solve it by factoring or using the quadratic formula.
However, upon further inspection, this equation cannot be factored easily. So, we will use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
For our equation, a = 1, b = -20, and c = 4:
x = (-(-20) ± √((-20)^2 - 4(1)(4))) / (2(1))
x = (20 ± √(400 - 16)) / 2
x = (20 ± √384) / 2
x = (20 ± 19.60) / 2
Using the positive square root:
x = (20 + 19.60) / 2
x = 39.60 / 2
x = 19.80
So, the solution to the equation Log(x^2 + 4) = 2 + log x - log 20 is x = 19.80.