Solve the logarithm equation
Log (x^2-5x+50)=2
10
To solve the logarithm equation, we need to get rid of the logarithm by converting it to exponential form.
The exponential form of logarithm equation is:
log(base a) x = b is equivalent to a^b = x
Using this, we can rewrite the given equation as:
10^2 = x^2 - 5x + 50
Simplifying the equation:
100 = x^2 - 5x + 50
Moving all terms to one side:
x^2 - 5x + 50 - 100 = 0
x^2 - 5x - 50 = 0
Now, we can solve this quadratic equation by factoring or using the quadratic formula.
Factoring:
(x - 10)(x + 5) = 0
Setting each factor to zero and solving for x:
x - 10 = 0 or x + 5 = 0
x = 10 or x = -5
So, the solutions to the logarithm equation are x = 10 or x = -5.
To solve the logarithm equation Log(x^2-5x+50) = 2, we need to rewrite it in exponential form.
In general, the exponential form of a logarithmic equation is given as:
y = b^x,
where y is the base, b is the logarithm base, and x is the exponent.
In this case, we have:
Log(x^2-5x+50) = 2,
which can be rewritten as:
x^2-5x+50 = 10^2.
Simplifying the right side of the equation gives:
x^2-5x+50 = 100.
Now, let's solve this equation. We can rearrange it to be a quadratic equation:
x^2 - 5x + 50 - 100 = 0,
x^2 - 5x - 50 = 0.
Notice that we can use the quadratic formula to find the solutions:
x = (-b ± √(b^2 - 4ac)) / (2a).
In this case, a = 1, b = -5, and c = -50. Substituting these values into the quadratic formula:
x = (-(-5) ± √((-5)^2 - 4(1)(-50))) / (2(1)),
x = (5 ± √(25 + 200)) / 2,
x = (5 ± √225) / 2,
x = (5 ± 15) / 2.
Thus, we have two possible solutions:
x = (5 + 15) / 2 = 20 / 2 = 10,
or
x = (5 - 15) / 2 = -10 / 2 = -5.
Therefore, the solutions to the equation Log(x^2-5x+50) = 2 are x = 10 and x = -5.
To solve the logarithm equation Log(x^2-5x+50)=2, we need to first understand the properties of logarithms.
In general, if we have an equation of the form Log(a) = b, then we can rewrite it as 10^b = a.
Applying this property to our equation Log(x^2-5x+50)=2, we can rewrite it as 10^2 = x^2-5x+50.
Simplifying the right side, we have 100 = x^2-5x+50.
To solve this quadratic equation, we rearrange it to get x^2 - 5x + (50-100) = 0.
Simplifying further, we get x^2 - 5x - 50 = 0.
Now, we can use factoring or the quadratic formula to solve for x.
Using factoring, we look for two numbers that multiply to give -50 and add to give -5. The numbers -10 and 5 satisfy these conditions. Therefore, we rewrite the equation as (x - 10)(x + 5) = 0.
Setting each factor equal to zero, we have x - 10 = 0 or x + 5 = 0.
Solving each equation, we find x = 10 or x = -5.
Therefore, the solution to the logarithm equation Log(x^2-5x+50)=2 is x = 10 or x = -5.