WEBVTT
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Calculate log base two of 192 minus log base two of three.
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This is an expression that we could just enter into our calculators.
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But actually, thereβs a very elegant solution.
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In general, the difference of the logarithms of two quantities to the same base is equal to the logarithm to that base of the quotient of the two quantities.
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The logarithms in our problem are to the same base.
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Theyβre both logarithms base two.
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And so their difference is the logarithm base two of their quotient, so log base two of 192 over three.
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This general formula is one of the so called laws of logarithms.
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And weβll come to prove it at the end of the video.
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192 divided by three is 64.
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And so we have log base two of 64.
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What is log base two of 64?
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If we call this π₯, then we can write this equation, which is in a logarithmic form, in exponential form as two to the power of π₯ equals 64.
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So really when Iβm asking for log base two of 64, Iβm asking two to the power of what equals 64.
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And hopefully we recognise 64 as a power of two.
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In fact, itβs the sixth power of two.
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Two to the power of six is 64.
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We see that π₯, which is log base two of 64 remember, is six.
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There we have it, no calculator required.
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We just use one of the laws of logarithms, which weβll now prove.
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This is the law that we want to prove written in terms of the arbitrary values π, π, and π.
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All the logarithms we have have base π.
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And so a relatively natural thing to do is to find π to the power of both sides.
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So we now have π to the power of this difference of logarithms is equal to π to the power of the logarithm of the quotient.
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On the left-hand side, we have π to the power of something minus something else.
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And we know from exponent laws that π to the power of something minus something else is equal to π to the power of that something divided by π to the power of that something else.
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So with something equal to log base π of π and something else equal to a log base π of π, we get that π to the power of log base π of π over π to the power of log base π of π is equal to π to the power of log base π of π over π.
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The right-hand side stays the same.
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We now have three things of the form π to the power of log base π of π₯ in our equation.
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What is log base π of π₯?
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Well if you remember from earlier, itβs the value that you have to raise π to the power of to get π₯.
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This is another rule that is really helpful to remember.
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π to the power of log base π of π₯ is equal to π₯.
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Applying this rule to our equation π to the power of log base π of π is π.
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π to the power of log base π of π is π.
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And π to the power of log base π of π over π is π over π.
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We have an equation which is clearly true.
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And if we follow the chain of logic in the other direction, we get the law of logarithms that we used to solve our problem.