WEBVTT
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Find the value of log base four of 1280 minus two times log base four of two minus log base four of five without using a calculator.
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So we just write out the expression again.
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To find the value of this expression as we are asked to do, weβre going to repeatedly simplify it.
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The first thing we can notice is that we can simplify this term: two times a log base four of two.
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We do this using the law of logarithms that π times log base π of π is equal to log base π of π to the power of π.
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And so two times log base four of two becomes log base four of two to the power of two.
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Essentially what weβve done is weβve moved the multiplicative constant, two, from the front of a log base four of two to become the power of the two inside the log.
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And of course two to the power of two, or two squared, is just four.
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And so we have log base four of 1280 minus log base four of four minus log base four of five.
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Now all three terms are just the logarithm or log base four of a number, and we can simplify this by using another law of logarithms.
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In general, log base π of π minus log base π of π is equal to a log base π of π over π.
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Applying this we see that log base four of 1280 minus log base four of four is log base four of 1280 over four.
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And of course, 1280 over four is just 320.
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And so our expression becomes log base four of 320 minus log base four of five.
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So now we have another difference of our logarithms to the same base; log base four of 320 minus log base four of five.
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And you might like to think what the next line of working will therefore be.
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The rule we use is log base π of π minus log base π of π is log base π of π over π.
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And so we get log base four of 320 over five.
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And of course that is log base four of 64.
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Are we done?
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Well, not quite.
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We can simplify further as it happens.
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Let π₯ equal log base four of 64.
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The exponential form of this relationship is four to the power of π₯ equals 64.
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Can you see what π₯ should be?
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Well, we know that four to the power of three, or four cubed, is 64.
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And so π₯ is equal to three.
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So we write that our answer is three.
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It might be surprising that we get such a nice answer from such a complicated expression.
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Two times log base four of two is one.
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But log base four of 1280 and log base four of five are both irrational numbers.
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And so something really nice had to happen for us to get an answer of three at the end.