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This is part 11 of log rhythms and we're going to be solving equations and bobbing logs
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and we're going to solve the following three equations in this video.
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Some of the things you're going to have to keep in mind as you're doing this is the meaning
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of log rhythms, how to change from a log to exponential form and that A is greater than
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0, B is greater than 0, both of these numbers must be greater than 0 and the base B cannot
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be equal to 1 and then these four properties are the most commonly used properties when
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you're solving logs.
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Alright we need to solve this.
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Try it on your own first.
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Okay, so I've got the log of 2x minus 3 base 5 equals 2 so I could just use the definition
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of logs, what does this mean?
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It means 5 to the second power equals 2x minus 3 or 2x minus 3 equals 5 squared, either
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way you want to write it.
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I like to write it this way so the variables on the left and so now this becomes a simple
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equation to solve, we just have to square 5 and we need to add 3 to both sides.
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So we get 2x is 28, divide both sides by 2 and x is 14.
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Now it's important when you're solving log equations that you plug the number back
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in and at least make sure that first of all the base is not negative or 0 or 1 which
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of course it doesn't exist as 5 and also what you're taking the log of this 2x minus
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3 also cannot be negative.
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So when you look back and you put in 14 for x you can see that this will not be a negative
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number but you can also completely check it by plugging it back into the original so
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we're going to check log base 5 of 2x minus 3 equals 2 and go ahead and really plug that
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number in for x.
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So we have log base 5 of 2 times 14 minus 3, I forgot to put a parentheses around that,
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you have to make sure that's a parentheses.
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So we have the log base 5 of 28 minus 3, well 28 minus 3 is 25 and this is something you
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can do in your head because the log of 25 base 5 means 5 to what x is equal to 25 and
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that will be 2.
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Now if it was something you couldn't do in your head you can get out the calculator
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and just do the log of 25 divided by the log of 5 and you should get 2 and so on the
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other side we also have 2.
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At the minimum if you don't go through an entire check make sure at least that you're
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not ever taking the log of a negative number.
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So for this problem the answer is 14.
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All right so try solving this one on your own.
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Notice it's not written as a single log on the left hand side so that will be your first
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step.
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All right so let's do that.
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Well we have the log of something plus the log of something else so we can use the product
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property that means it's the log of x times x plus 21.
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Now notice that only works if the base is the same and notice you don't see the base.
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So if there's no base shown remember that means the base is 10.
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So if you want you could write the 10 that's optional but if you don't write it you have
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to at least understand that's what's being assumed.
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So now we've got the log of something base 10 equals 2 so that's something will be equal
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to 10 squared.
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So we have x times x plus 21 equals 10 squared.
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We get out of log form and exponential form and now we need to solve this equation.
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Well first I'm going to have to distribute on the left and on the right side we need
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to square 10 and now when I analyze this I see that I have a quadratic equation.
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Well to solve a quadratic equation you set the equation equal to 0 and if you can factor
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your factor and if you can't factor then you could use the quadratic equation or you
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can use the square root property you know complete the square.
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But how convenient this actually factors.
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So this is going to be x plus 25 times x minus 4 equals 0 so we set each factor equal
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to 0.
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So we get two possibilities x can be negative 25 or x can be 4.
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Now remember when you you must check both of these in the original.
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So if we go back up to the original up here if you plug in negative 25 for x you're
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taking the log of a negative number that's not allowed.
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So you could go through the trouble of writing it down but hopefully right away you see
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that that's not going to be a solution.
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It's not that x can't be negative it's that when you're checking it back in the original
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you can't take the log of a negative number.
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All right now if you put in 4 for x we'll be okay.
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If we put in 4 here yes because we're taking the log of a positive and if you put in
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4 over here you have 4 plus 21 also positive.
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So 4 might work and that's our next step is to actually check it completely in the
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original.
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All right so let's plug in 4 for x.
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We have the log of 4 plus the log of 4 plus 21.
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So that's the log of 4 plus the log of 25.
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Now there's two ways you can go from here.
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You could use your calculator and put in the log of 4 plus the log of 25 and add it together
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and you should get the number 2 if you're careful.
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Okay but if not you could use your property of logs right here and you have the log of
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remember what this means you multiply 4 times 25.
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So that would be the log of 100 and then remember even though I didn't write it these
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are all base 10.
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So the log of 100 base 10 will tend to what power is 100?
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2.
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So then I can see that I have 2 on the left hand side, I have 2 on the right hand side
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so it checks.
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So again if you want to use your calculator you could also check it that way you'll
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put in log of 4 and then you'll add it to log of 25.
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Now if you write down the approximations your calculator it will not come out exactly
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correct because there's some rounding errors that might occur but if you don't round
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and you do everything in your calculator should come out to 2 probably.
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So what does this tell me?
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This tells me that x equals 4 is the solution.
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Alright see if you can solve this one on your own by putting the video on pause and
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trying it first.
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Okay so again we don't have a single log on the left hand side but we have a subtraction
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so we can use the quotient property.
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The log base 3 of n over 4 is equal to 2.
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If you want you can put parentheses around that, it's up to you.
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So the 3 to the second power 3 squared equals n over 4 or n over 4 equals 3 squared.
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So we have n over 4 equals 9 and then you can just multiply both sides by 4 to solve
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for n and we got 36.
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Now you want to go back up to the original and make sure this makes sense.
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So first of all when you put n and make sure you're not taking the log of a negative number
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and you're not because it will be positive there and now let's do the full check.
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So let's put in 36 for n, log of 36 base 3 minus the log of 4 base 3.
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So now again you could do this in your calculator, you could compute the log of 36 base 3 which
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happens to be the log of 36 over the log of 3 or what I think would be easier is to simply
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use your property of logs here, the log base 3 of that will be 36 over 4, which is 9.
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And then what is the log of 9 base 3 that you could do in your head?
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3 to what power would be 9?
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That would be 2.
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So you have 2 on both sides.
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So that means 36 is the correct solution to that problem.