Question Video: Solving a Logarithmic Equation Where the Argument of the Logarithm Is a Quadratic Equation Mathematics

Solve the equation log(𝑥² − 𝑥 − 2) = 1, where 𝑥 ∈ ℝ.

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Video Transcript

Solve the equation log of 𝑥 squared minus 𝑥 minus two equals one, where 𝑥 is in the set of real numbers.

The first thing to note in this question is that when we see log on its own, what we actually mean is log to the base 10. And this is gonna come in very useful when we trying to solve this equation. Well, what we’re going to do now is we’re going to use one of our logarithm rules to help us rewrite our equation. And that is, if we have log to the base 𝑎 of 𝑎, this is equal to one so long as 𝑎 is positive and not equal to one. So therefore, what we can say is that log of 𝑥 squared minus 𝑥 minus two is equal to the log of 10. And that’s because, as we’ve already said, if we just have log on its own, it’s the same as log to the base 10, so log to the base 10 of 10 would just be one.

Well, now, as we’ve got the same log on each side of the equation to the same base because they’re both log to the base 10, what we can do is just take a look at our argument. And we can look at the arguments on both sides of the equation and actually equate these. So therefore, we can say that 𝑥 squared minus 𝑥 minus two equals 10. So then what we can do is subtract 10 from each side of the equation. So then we’ve got a quadratic 𝑥 squared minus 𝑥 minus 12. And then we have it equal to zero. Now, in order to solve this quadratic, we’ve got many different methods we can use. However, the most straightforward method is going to be to factor.

And then just to remind ourselves how to factor, what we want to do is find two factors whose product is the numerical value, so in this case negative 12, and whose sum is the coefficient of 𝑥, so in this case negative one. I’ve just shown that here in orange because obviously when you write negative one 𝑥, you would just have negative 𝑥. So therefore, we’re gonna get 𝑥 minus four multiplied by 𝑥 plus three. And that’s because negative four multiplied by positive three is negative 12 and negative four add three is negative one. So what we’re gonna get is the results for 𝑥 of four or negative three. And we get that because what we’re looking for are 𝑥-values that are gonna make our parentheses equal to zero.

So, for example, if we had four minus four, this would be equal to zero. And the reason we want to do that is because we want the left side to equal zero, and zero multiplied by anything is zero. And that’s because we want it to be the same as the right side, which is zero. So therefore, what we can say is that the solutions to the equation log of 𝑥 squared minus 𝑥 minus two equals one, where 𝑥 is in the set of real numbers, are negative three and four.