Hey, everyone. Whenever we solve polynomial equations like x3=216, we can simply do the reverse operation in order to isolate x. So here, since x is being cubed, I could simply take the cube root of both sides to cancel that, isolating x, giving me my answer that x=216. But what if my variable x is instead in my exponent, like this 2x=8? How are we going to isolate x here?
Well, I know that looking at this, I can just think, okay, how many times does 2 need to be multiplied by itself to give me 8? And I know that my answer is simply x is equal to 3. But what if I'm given something like 2x=216? I really don't want to have to multiply 2 enough times to get up to 216. So is there not just an operation I can do to cancel out that 2 and leave me with x?
Well, here I'm going to show you that there is an operation that does just that because the reverse or inverse of an exponential is actually taking the logarithm or a log. Now, the first time you work with logs, they can be a bit overwhelming. But here I'm going to walk you through exactly what a log is and how we can use it to actually make our lives easier, especially when working with exponents. So now that we know that the reverse or inverse operation of an exponential is simply taking the log, we can go ahead and take the log of both sides of this equation in order to isolate x. So I take log 2(2x), and that's equal to log(216).
But we actually do need to consider one more thing here because whenever we cancelled the 3 on our x3, we took the cube root. We didn't take the square root or the 4th root or anything else. We took the cube root to cancel that 3. And we need to consider something similar when working with logs because logs and exponentials need to have the same base as each other to cancel. So here, since I have an exponential of base 2, I want my log to have a base of 2 as well.
So really, I want to take a log base 2 of 2x, and then on the other side, a log base 2 of 216. Now that my log and my exponential have the same base, then it's going to fully cancel out, leaving me with just x=log 2(216). Now it's fine to leave it in this form here. This is actually called our logarithmic form. And we're later going to learn how to fully evaluate these and get a number, but for now, we're just going to keep it in that log form.
Now that we're here, what exactly does this statement mean? We have log 2(216). Well, a log is actually giving us the power that some base must be raised to in order to equal a particular number. But what does all of that mean? Well, looking at our function here or our equation here, log 2(216), this is really saying, okay, what power does 2 need to be raised to in order to give me 216?
Now, this statement here I mentioned is in its logarithmic form, and it's actually an equivalent statement to our very first equation, 2x=216. This is just in its exponential form, and we basically translated it into its logarithmic form. Now we're going to have to do this for multiple statements. We're going to have to translate and convert expressions between these two forms. So diving a bit deeper in converting between these two forms, let's start by taking this equation in its exponential form, 3x=81, and putting it into log form.
Now whenever we convert between these two forms, no matter what we're going to or from, we're always going to start at the same place. We're going to start with our base. So this base 3 of our exponent is going to become the base of our log.
