Log3 (x^2-5x+9)=1
To solve the equation Log3 (x^2-5x+9)=1, first, we need to get rid of the logarithm. We can do that by recognizing that if Log3 (x^2-5x+9)=1, then 3^1 = x^2-5x+9.
Now, the equation becomes:
3^1 = x^2-5x+9
Simplify the left side:
3 = x^2-5x+9
Now, solve for x by first moving all the terms to one side:
x^2 - 5x + 6 = 0
Factor the quadratic equation:
(x - 2)(x - 3) = 0
Now we have two possible solutions:
x - 2 = 0 => x = 2
x - 3 = 0 => x = 3
So the two solutions are x=2 and x=3.
To solve the equation log3 (x^2 - 5x + 9) = 1, we need to isolate x.
First, we'll rewrite the equation in exponential form. In logarithmic form, log3 (x^2 - 5x + 9) = 1 means that 3^1 = x^2 - 5x + 9.
Simplifying the exponential equation, we have 3 = x^2 - 5x + 9.
Rearranging the terms to bring everything to one side, we get x^2 - 5x + 6 = 0.
Now, we need to factor the quadratic expression or use the quadratic formula to solve for x. Factoring is usually the easiest approach when possible.
To factor, we look for two numbers whose product is 6 and whose sum is -5 (the coefficient of the x term in our quadratic expression). The numbers that satisfy these conditions are -2 and -3.
Hence, we can rewrite the quadratic expression as (x - 2)(x - 3) = 0.
Setting each factor equal to zero, we have two equations:
x - 2 = 0 or x - 3 = 0.
Solving each equation separately, we find:
For x - 2 = 0, when we add 2 to both sides, we get x = 2.
For x - 3 = 0, when we add 3 to both sides, we get x = 3.
Therefore, the solution to the given equation log3 (x^2 - 5x + 9) = 1 is:
x = 2 or x = 3.
To solve the equation log₃(x² - 5x + 9) = 1, we need to eliminate the logarithm first by rewriting the equation in exponential form.
In logarithmic form, logₐ(b) = c, the base "a" raised to the power of "c" equals "b".
So, we can rewrite the equation log₃(x² - 5x + 9) = 1 as 3¹ = x² - 5x + 9.
Simplifying, we have:
3 = x² - 5x + 9
Now, we can rearrange the equation to the quadratic form:
x² - 5x + 9 - 3 = 0
x² - 5x + 6 = 0
Next, we need to factorize the quadratic equation:
(x - 2)(x - 3) = 0
Setting each factor to zero, we have two equations:
x - 2 = 0 or x - 3 = 0
Solving each equation separately:
For x - 2 = 0, adding 2 to both sides gives:
x = 2
For x - 3 = 0, adding 3 to both sides gives:
x = 3
Therefore, the solutions to the equation log₃(x² - 5x + 9) = 1 are x = 2 and x = 3.