WEBVTT
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Consider the function ๐ of ๐ฅ equals log to the base eight ๐ฅ.
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Find the value of ๐ of two.
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Well, to solve this problem, what weโre gonna do is change our expression from log form into exponent form.
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And we could do that using a relationship we know.
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If we have ๐ฆ is equal to log to the base ๐ of ๐ฅ, then if we rewrite this in exponent form, weโre gonna get ๐ฅ is equal to ๐ raised to the power of ๐ฆ.
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Okay, great.
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So, weโve got this little relationship that we can use.
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Iโm gonna use it on the expression that we have.
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But first of all, what weโre gonna do is substitute in two for our value of ๐ฅ cause weโre looking for the value of ๐ of two.
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So, if we look at what weโve got, weโve got ๐ฆ is equal to log to the base eight of two.
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And thatโs because Iโve substituted in two for ๐ฅ.
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And Iโve also called our ๐ of ๐ฅ ๐ฆ because thatโs interchangeable.
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So now, what we can do is we can change this into exponent form.
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And we can do that because we know that our ๐ is going to be eight because thatโs our base, and our ๐ฅ is going to be two.
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So therefore, we can say that two is gonna be equal to eight raised to the power of ๐ฆ.
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Well, to be able to solve this problem, what we want to do is think about having two values that have the same base.
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Well, letโs think about eight.
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Can eight be written as two raised to a power?
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Well, we know that two multiplied by two multiplied by two is equal to eight.
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So therefore, two raised to the power of three is equal to eight.
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So, that means we can rewrite eight as two raised to the power of three.
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So then, what weโre gonna have is two equals two raised to the power of three.
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And this is to the power of ๐ฆ.
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So then, what weโre gonna do is utilize one of our exponent rules.
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And that is if we have ๐ฅ raised to the power of ๐, then this is raised to the power of ๐, that is the same as ๐ฅ raised to the power of ๐๐.
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So, you multiply the exponents.
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So therefore, we can say that two is equal to two raised to the power of three ๐ฆ.
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Well, we donโt usually write it or include it but two is, in fact, two raised to the power of one.
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So, we can now see that weโve got the base two on the left- and right-hand side of our equation.
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That means what we can do is we can equate our exponents.
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So, what we can say is that one is equal three ๐ฆ.
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So then, if we divide both sides of our equation by three, what weโre gonna get is a third is equal to ๐ฆ.
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So, weโve solved the problem.
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And we can say that if the function ๐ of ๐ฅ equals log to the base eight of ๐ฅ, then the value of ๐ of two is gonna be equal to one-third.