solve the logarithm equation log2(x^2-2x+5)=2
log2(x^2-2x+5)=2
x^2 - 2x + 5 = 2^2 by definition
x^2 - 2x + 1 = 0
(x - 1)^2 = 0
x - 1 = 0
x = 1
To solve the equation log2(x^2 - 2x + 5) = 2, we can apply the definition of logarithms. In this case, the base is 2 and the logarithm is equal to 2.
Step 1: Rewrite the equation in exponential form.
In exponential form, the equation log2(x^2 - 2x + 5) = 2 can be written as 2^2 = x^2 - 2x + 5.
Step 2: Simplify the equation.
2^2 simplifies to 4, so the equation becomes 4 = x^2 - 2x + 5.
Step 3: Rearrange the equation.
Move all terms to one side to bring the equation to the form of a quadratic equation. Subtracting 4 from both sides, we get x^2 - 2x + 5 - 4 = 0, which simplifies to x^2 - 2x + 1 = 0.
Step 4: Solve the quadratic equation.
To solve the quadratic equation x^2 - 2x + 1 = 0, we can use the quadratic formula: x = (-b ± √(b^2 - 4ac)) / (2a).
For this equation, a = 1, b = -2, and c = 1.
Plugging these values into the quadratic formula, we get:
x = (-(-2) ± √((-2)^2 - 4(1)(1))) / (2(1))
x = (2 ± √(4 - 4)) / 2
x = (2 ± √0) / 2
x = (2 ± 0) / 2
x = 2 / 2
x = 1
Step 5: Check the solution.
Let's substitute x = 1 back into the original equation to verify if it satisfies the equation.
log2(1^2 - 2(1) + 5) = 2
log2(1 - 2 + 5) = 2
log2(1 + 3) = 2
log2(4) = 2
The logarithm base 2 of 4 is indeed equal to 2, so our solution x = 1 is valid.
Therefore, the solution to the logarithmic equation log2(x^2 - 2x + 5) = 2 is x = 1.
To solve the logarithm equation log2(x^2-2x+5)=2, follow these steps:
Step 1: Rewrite the equation using the logarithmic property.
According to the logarithmic property, loga(b) = c is equivalent to a^c = b.
Applying this property to the equation log2(x^2-2x+5) = 2, we get 2^2 = x^2-2x+5.
Step 2: Simplify the equation.
Simplify the left side of the equation: 2^2 = 4.
On the right side, combine like terms: x^2-2x+5.
Step 3: Set the equation equal to zero.
Set the equation x^2-2x+5 = 4 equal to zero: x^2-2x+5 - 4 = 0.
This gives us the quadratic equation x^2-2x+1 = 0.
Step 4: Solve the quadratic equation.
To solve the quadratic equation x^2-2x+1 = 0, factor it: (x-1)(x-1) = 0.
This means that (x-1) = 0 or (x-1) = 0.
Therefore, x = 1.
Step 5: Check the solution.
Substitute x = 1 back into the original equation log2(x^2-2x+5) = 2,
log2((1)^2-2(1)+5) = 2.
Simplify the right side: log2(4) = 2.
Since 2^2 = 4, the equation is true.
Therefore, the solution to the logarithm equation log2(x^2-2x+5)=2 is x = 1.