What is the solution set of the equation log to the base 𝑥 of nine 𝑥 minus 18 equals two?
So to help us solve this, what we can use is some of our relationships for logarithms. So, if we had log to the base 𝑏 of 𝑥 equals 𝑘, well, then what we can say is that 𝑥 is equal to 𝑏 to the power of 𝑘. So what we’re gonna do now is take a look at our equation and see how we can change it into exponent form. So first of all, what we’re gonna do is take a look at the corresponding parts from our equation to the relationship we looked at. So we’ve got the 𝑏 in the relationship is gonna be our 𝑥. The 𝑥 is gonna be our nine 𝑥 minus 18. And the 𝑘 is gonna be our two. And we can do this because our equation is in that form, log to the base 𝑏 of 𝑥 equals 𝑘.
So then what we can do is actually convert this into our exponent form. So, what we’re gonna get is nine 𝑥 minus 18 is equal to 𝑥 squared. Then what we can do is rearrange this to make it a quadratic that we know how to solve. So, if we subtract nine 𝑥 and add 18 to both sides of the equation, we’re gonna get zero is equal to 𝑥 squared minus nine 𝑥 plus 18. And this is actually a nice, straightforward quadratic for us to solve because what we can do is use factoring to solve it.
So now, in order to actually factor our quadratic, what we need to do is find two values whose product is positive 18 and whose sum is negative nine. Well, these are negative three and negative six. So we get zero is equal to 𝑥 minus three multiplied by 𝑥 minus six. And then to check why we’ve got that, we do negative three multiplied by negative six, which is positive 18, and negative three add negative six gives us negative nine. So therefore, our 𝑥-values are gonna be three or six. And that’s cause these are the values that make our parentheses equal to zero. So, for instance, if we have three minus three, this is zero, and zero multiplied by anything is just zero. We want that on the right-hand side because the left-hand side is zero.
So therefore, we can say that the solution set of the equation log to the base 𝑥 of nine 𝑥 minus 18 equals two is three and six.