Question
1/2log 8 base 4 + log 32 base 4 - log 2 base 4
Log base 10(x^2- 4x + 7)=2
Log base 10(x^2- 4x + 7)=2
Answers
GPT 3.5
To solve the equation log base 10(x^2 - 4x + 7) = 2, we can rewrite it in exponential form using the definition of logarithms:
10^2 = x^2 - 4x + 7
Simplifying, we have:
100 = x^2 - 4x + 7
Rearranging the equation to the standard quadratic form, we get:
x^2 - 4x + 7 - 100 = 0
Simplifying further, we have:
x^2 - 4x - 93 = 0
To factor this quadratic equation, we need to find two numbers that multiply to give -93 and add up to -4. The numbers are -9 and 11. Therefore, the factored form of the equation is:
(x - 9)(x + 11) = 0
Setting each factor equal to zero, we have:
x - 9 = 0 or x + 11 = 0
Solving these equations, we get:
x = 9 or x = -11
Hence, the solutions to the equation are x = 9 and x = -11.
10^2 = x^2 - 4x + 7
Simplifying, we have:
100 = x^2 - 4x + 7
Rearranging the equation to the standard quadratic form, we get:
x^2 - 4x + 7 - 100 = 0
Simplifying further, we have:
x^2 - 4x - 93 = 0
To factor this quadratic equation, we need to find two numbers that multiply to give -93 and add up to -4. The numbers are -9 and 11. Therefore, the factored form of the equation is:
(x - 9)(x + 11) = 0
Setting each factor equal to zero, we have:
x - 9 = 0 or x + 11 = 0
Solving these equations, we get:
x = 9 or x = -11
Hence, the solutions to the equation are x = 9 and x = -11.