WEBVTT
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Determine the solution set of the equation log to base eight of π₯ minus six plus log to base eight π₯ plus six equals log to base eight 64 in the set of real numbers.
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To enable us to begin to solve this problem and actually solve the equation, all we actually want to do is apply a log law.
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And the log law that weβre gonna be using is that log to base π of ππ, so thatβs π multiplied by π, is equal to log to base π of π plus log to base π of π.
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So therefore, if we look back at our equation, we can see, well, actually they all have the same base cause the eight is our base.
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Itβs the same as our π in our log law.
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So therefore, we can say that our log to base eight π₯ minus six plus log to base eight π₯ plus six is equal to log to base eight of 64 is equal to π₯ minus six multiplied by π₯ plus six is equal to 64.
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And thatβs because if we look across to our log law, we can see that well the π₯ minus six is like our π.
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And the π₯ plus six is like our π.
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And if we actually multiply them together, thatβs gonna be equal to our 64, which was our ππ.
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And now we can solve this equation.
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And the first thing we need to do is expand the parentheses.
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So we get π₯ squared plus six π₯ minus six π₯ minus 36 is equal to 64.
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And then we can actually collect our terms here.
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A key thing here as well to have a look at is the fact that these two parentheses are actually the difference of two squares.
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So what it means is that, actually, our π₯ terms are gonna cancel each other out.
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So weβre just gonna be left with our π₯ squared minus 36.
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And this is equal to 64.
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Again, we say they cancel each other out.
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What that means is because if we have plus six π₯ and then minus six π₯, thatβs just zero.
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Okay, so now our next step.
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Well, for our next step, we actually add 36.
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So therefore, we get π₯ squared is equal to 100.
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And then if we root each side, we get π₯ is equal to 10.
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So therefore, we can actually say that the solution set of the equation log to base eight of π₯ minus six plus log to base eight π₯ plus six is equal to log to base eight 64 is 10.