A logarithmic equation can have at most one extraneous solution.
Can a logarithmic equation have a negative solution?
Logarithms cannot have non-positive arguments (that is, arguments which are negative or zero), but quadratics and other equations can have negative solutions. Each log in the equation had the same base, and each side of the log equation ended up with the value, so the solution “checks”.
Why are there extraneous solutions?
The reason extraneous solutions exist is because some operations produce ‘extra’ answers, and sometimes, these operations are a part of the path to solving the problem. When we get these ‘extra’ answers, they usually don’t work when we try to plug them back into the original problem.
When can you apply the 1 1 property of logarithms to solve an equation?
The one-to-one property can be used if both sides of the equation can be rewritten as a single logarithm with the same base. If so, the arguments can be set equal to each other, and the resulting equation can be solved algebraically.
What is the natural logarithmic function of 0?
What is the natural logarithm of zero? ln(0) = ? The real natural logarithm function ln(x) is defined only for x>0. So the natural logarithm of zero is undefined.
How do you check for extraneous solutions?
To find whether your solutions are extraneous or not, you need to plug each of them back in to your given equation and see if they work. It’s a very annoying process sometimes, but if employed properly can save you much grief on tests or quizzes.
How do you know if a solution is extraneous or extraneous?
To determine if a solution is extraneous, we simply plug the solution into the original equation. If it makes a true statement, then it is not an…
Why do we have extraneous solutions in logarithms?
Lastly, extraneous solutions when dealing with logarithms are simply due to your lack of understanding of how complex numbers play into logarithms. When you must use the definition that a logarithm is only defined for positive real input, then you will get extraneous solutions for the very reason that you have that parameter in place.
Why are some solutions extraneous to the original equation?
Thus, a solution may be extraneous because it results from using a negative square root instead of the principle square root. Here’s an example: To solve this, we square both sides: Rearrange, simplify, factor, solve: Now, plug 9 into the original equation, and you’ll see that it does work. But what happens when you plug in 4?
How can you tell if a solution is an extraneous root?
To put it another way, an extraneous root will prove to be extraneous only by making a denominator of the original equation zero. So all you need to do to test for an extraneous root is to make sure each solution is in the domain of the equation — that is, none makes the denominator zero. In this case, it is clear that they don’t.
Why do extraneous solutions occur in radical notation?
However, when we write radical notation, we are, be definition, referring to the principle square root, which is the positive value. Thus, a solution may be extraneous because it results from using a negative square root instead of the principle square root. Here’s an example: To solve this, we square both sides:
