SOLUTION: Consider the function f(x) =log2(2/x^2-3x+2). For what value(s) does f(x)=0

Question 818511: Consider the function f(x) =log2(2/x^2-3x+2). For what value(s) does f(x)=0
Answer by jsmallt9(3758) (Show Source): You can put this solution on YOUR website!
First of all, I'm assuming the 2 next to the log is an indication of the base of the log. In the future either use some English to make it clear (like "base 2 log of (2/(x^2-3x+2))" or teach yourself algebra.com's formula syntax. (Click on the "Show source" link above to see what I typed (and what you could type) to make the logarithms look good.)

Second, I'm assuming that the function is:

and not:

which is what you posted. If I am right then please put parentheses around multiple-term numerators, denominators, exponents, function arguments, etc. so that the expression means what you intend it to mean.

Third, please post your questions in an appropriate category. You posted this under Polynomials. But this problem has a lot more to do with logarithms than polynomials. Posting your question in an appropriate category will increase your chances of a speedy response.

f(x) = 0
Replacing f(x) with 0:

Next we rewrite this in exponential form:

which simplifies to:

Now that the variable is out of the logarithm we can solve the equation. Multiplying both sides by (to eliminate the fraction) we get:

This is a quadratic equation so we want one side to be zero. Subtracting 2 from each side we get:

Now we factor:

Next the Zero Product Property:
x = 0 or x-3 = 0
Solving the second equation we get:
x = 0 or x = 3

Last we check. This is not optional! A check must be made to ensure that all bases and arguments of all logs are valid. (Valid bases are positive but not 1 and valid arguments are positive.) If a "solution" makes any base or argument invalid it must be rejected.

Use the original equation to check:

Checking x = 0:

Simplifying:

We can now see that the base, 3, and the argument, 2/2 (or 1), are both valid. So this solution passes the check.

Checking x = 3:

Simplifying:

Again this solution checks out.

So there are two solutions to your equation: x = 0 or x = 3.
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