Hey everyone. We just learned that whenever we have an exponential equation like 16 is equal to 2 to the power of x, we can simply rewrite each side to have the same base and then set our powers equal to each other in order to get our final answer. But what if we're given an exponential equation like 17 is equal to 2 to the power of x? It's not exactly easy to rewrite 17 as a power of 2. And don't worry, you're not going to have to figure out what crazy decimal that 2 needs to be raised to in order to get 17 because here we can simply solve this equation using logs. So here I'm going to walk you through how we can just take the natural log of each side and use properties of logs that we already know in order to get our final answer. So let's go ahead and get started.
Now let's just walk through an example together and just go through the steps that we're going to need to follow in order to get our answer here for each of these exponential equations. So for our first equation, we have 10x+64=100. So starting with step 1, the very first thing we want to do is going to be to isolate our exponential expression. So here our exponential expression is this 10x. So I want that by itself on one side of my equation. So in order to do that, I need to go ahead and move this 64 over which I can do by subtracting 64 from each side. Now when I do that, I'm going to end up with 10x=36. Now from here, step 1 is done, and I can go ahead and move on to step 2.
Now step 2 is when we're going to determine what log we should take. We're either going to take the natural log or the common log, and here is how we choose. If it has a base of 10, if our exponential expression has a base of 10, which here I have 10x, I'm going to take the common log, just log of both sides. Now if instead it did not have a base of 10, I would instead take the natural log. These are the only two logs you'll use whenever solving exponential equations. So here, like I said, we have a base of 10 so I can go ahead and just take the common log of both sides. So doing that I get log of 10x and that's equal to log of 36. Remember, whatever you do to one side, you have to do to the other. So we are, of course, taking the log of both sides. Now step 2 is done. And we can go ahead and move on to step 3, where we're going to use our log rules in order to get x out of our exponent. So here, x is in my exponent. But I know that using my power rule, I can go ahead and move that over. So moving that x to the front here, I end up with x times log of 10. And that's equal to log of 36. Now step 3 is done. We have x out of our exponent. And now we can move on to step 4 and go ahead and solve for x. Now looking at this equation, I have x times log of 10 is equal to log of 36. And you may notice here that I have this log of 10. And since log is just log base 10, I know that this is just going to end up being 1. So this is really just x times 1, which is just x. So here I end up with x, and that's equal to log of 36. Now here we could be done. X is equal to the log of 36. This is an acceptable answer if you're not asked to approximate because the log or natural log of some number is a constant. So we really don't need to do anything else from here unless we're explicitly asked to. So step 4 is done. But if you're asked to approximate, we can go ahead and plug this into our calculator.
Now let's move on to our next example. We have 3 is equal to 2x+1. Now we want to restart our steps starting from step 1. And we want to go ahead and isolate our exponential expression. Now our exponential expression is actually already by itself here, so I don't need to do anything else. Step 1 is done. I can move on to step 2. Now in step 2, we're either going to take the log or natural log. And here, I do not have a base of 10, so I'm going to go ahead and take the natural log of both sides. So here I take natural log of 3. And that's equal to the natural log of 2x+1. Now step 2 is done. And I can go ahead and move on to step 3 and use my log rules to take x out of my exponent. Now again, here, I'm going to use the power rule. So I'm going to take this exponent up by 2 and pull it to the front of that natural log. So on this left side, I still have the natural log of 3, But now my right side has become x+one times the natural log of 2. Make sure you pull your entire exponent when using the power rule. So from here, I've taken x out of my exponent. Step 3 is done. Moving on to step 4, we want to go ahead and solve for x. So this looks a little bit crazy here, but we can simplify this significantly and get x all by itself. So here I have this natural log of 3, and that's equal to x+1 times the natural log of 2. But remember that the log or natural log of something is just a constant, and I can treat it just like I would any other number. So I can go ahead and move this natural log to the other side by simply dividing by it like I would any constant. So dividing both sides by the natural log of 2, I end up with the natural log of 3 divided by the natural log of 2, and that's equal to whatever I have left over here, which happens to be x plus 1. Now I have one final step here in isolating x, getting it by itself, and that is subtracting 1 from both sides. So I am left with that canceling. And I have my final answer here that the natural log of 3 divided by the natural log of 2 minus 1 is equal to x. Now this looks a little bit scary, but remember, these natural logs are just numbers. So this is really just a bunch of constants, natural log of 3, natural log of 2, and 1. So this is an acceptable answer. But of course, if you're asked to approximate using a calculator, you can go ahead and plug this into your calculator to get a decimal. Now whenever you do that, you're going to get a final answer of x is equal to 0.58. And we are completely done here. We have our final answer. Now that we know how to solve exponential equations using logs as well as using like bases, We can solve any exponential equation that gets thrown at us. Thanks for watching, and let me know if you have any questions.