Hey, everyone. We just solved a bunch of different exponential equations. But what about logarithmic equations? Well, now that we're faced with solving yet another new type of equation, you may be worried yet again that we're going to have to learn something brand new in order to solve these. But, again, you don't have to worry about that at all because here I'm going to show you that there are only ever going to be 2 types of log equations that you run into, 2 logs of the same base that are set equal to each other or a single log set equal to a constant. And both of them boil down to solving a basic linear equation like we've done a million times before. So let's go ahead and just jump right in.
Now with exponential equations, we saw that whenever we had exponents of the same base, we could simply take those powers and set them equal to each other in order to get our answer. And we can actually do the same exact thing whenever working with log equations. So here we have these log base 2. And since these have the same exact base, I can simply take what we're taking the log of, so x+5, and go ahead and set those equal to each other. So I'm left with just a basic linear equation that I can then solve to get x is equal to 4.
So whenever we have 2 logs of the same base, we can just take what we're taking the log of and set them equal to each other and solve for x.
So let's look at a slightly more complicated example here. We have the natural log of (x+4) minus the natural log of 2, and that's equal to the natural log of 8. Now here, it might not be immediately obvious that we're going to be able to just set some stuff equal to each other. So let's take a closer look here. I know that these are all natural logs. So they are all logs of base e, that same base. And here, I can actually use my log properties in order to condense this a little bit more. So I can actually go ahead and use my quotient rule here since I have this subtraction happening. So I can go ahead and condense this into the natural log of (x+2), and that's equal to the natural log of 8. Now that these have the same base, I have this natural log on both sides, I can go ahead and set (x+2) equal to 8 in order to get the linear equation that I can go ahead and solve for x with. So this becomes x+2 and that's equal to 8. Now we can just solve this basic linear equation by isolating x. So my first step here is going to be to multiply both sides by 2, and that will cancel leaving me with x+4, and that's equal to 8 times 2 which is 16. Now one final step in isolating x here is going to be to subtract 4 from both sides, leaving me with x is equal to 12. Now that's my final answer, and I'm done.
All I had to do was make sure that both sides had the same base and then go ahead and set things equal to each other, giving me a basic linear equation. Now let's look at our other type of log equation. If I cannot rewrite it with like bases and I actually just have a single log, like, say, log base 2 of (4x), and that's equal to a constant. So here you may be worried that this is where it's going to be complicated, but it's not because we're simply going to put this in exponential form and then solve from there. So let's go ahead and work through this example together.
So we have log base 2 of (4x) and that's equal to 5. Now our very first step here is going to be to isolate our log expression, which here is log base 2 of (4x). So it's already isolated and I'm already done with step 1. I can move on to step 2, which is the meat of this problem, putting it in exponential form. Now remember when putting things in exponential form, we're always going to start with that base. So here we're going to start with that 2, and then circle to the other side of our equal sign. So raise that 2 to the power of 5 and then come back to the other side of our equal sign and set that equal to 4x. So we've completed step 2. This is in its exponential form. We can go ahead and move on to step 3 and actually solve for x. So here I can go ahead and simplify this exponent, this 25, I know that if I multiply 2 by itself 5 times, I'm going to end up with 32. So this is 32, and that's equal to 4x. Now here to isolate x, I can just divide both sides by 4, canceling on that side, leaving me with my answer that x is equal to 8. Now here I already have my answer. I've completed step 3. But we actually have to perform one final check here because we actually want to go ahead to our very last step and check our solution by plugging x into m.
Now m is just what we are taking the log of. So in this case, it's going to be this 4x here. So we wanna plug our answer into that 4x. And in this case, since we got an answer of 8, we're gonna plug in 8 for x. So I'm going to get 4 times 8, which is simply 32. And the thing we're looking for here is the sign of this number, whether it's positive or negative. So here if m is greater than 0, if it's positive, we are done and this is our solution. So in this case, our solution is good and I know that x is equal to 8. But if I got a negative number here, this would actually not be a solution at all. So remember here you cannot take the log of a negative number. It is not a solution if you do get a negative number that you're taking the log of.
So now that we know how to solve all of these log equations and we already knew how to solve all of our exponential equations, we are good to go. And let's get some more practice. Let me know if you have questions.