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INSTRUCTOR: In this video, we're briefly going to go over

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some properties of logs that you need to be familiar with.

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The first one is the power rule.

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Let's say if you have log A raised to the N.

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You can move the exponent in front.

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This is equal to N log A.

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The second is the product rule.

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Log A times log B is equal to log A plus log B.

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Then you have the quotient rule.

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Log A divided by log B is log A minus log B.

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We can use this in order to evaluate logs.

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For instance, let's say if we want to simplify log base 5 that's 5 raised to 7.

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What we can do is move the 7 to the front.

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This is equal to 7 log base 5 of 5.

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A log base 5 of 5 because they're the same,

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it's equal to 1 and 7 times 1 is 7.

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Let's try another example like that.

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Log base 2 of 8 raised to fifth power.

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Let's move the 5 to the front.

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This is equal to 5 times log base 2 of 8.

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Now, how many 2s do we need to multiply to get to 8?

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We need to multiply 3 2s, 2 to the third is 8,

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so log base 2 of 8 is 3,

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and 5 times 3 is 15.

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Now let's work on this example.

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Log base 2 of 16 times 8.

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We can use the product rule to separate the 16 and the 8.

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If you recall, log of A times B equals log A plus log B.

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Therefore, log base 2 of 16 times 8 is going to be log base 2

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of 16 plus log base 2 of 8.

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Now, log base 2 of 16,

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that's 4 because 2 to the fourth power is 16.

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Log base 2 of 8 is 3.

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The final answer is 4 plus 3 which is 7.

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Try this one. Log base 3 of 27 times 81.

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First, let's separate it into two logs.

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This is log base 3 of 27 plus log base 3 of 81.

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3 to the third power is 27, so log base 3 of 27 is 3.

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Log base 3 of 81 is 4 because 3 to the fourth power is 81,

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and this adds to 7. What about division?

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What is log base 4 of 256 minus 64?

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Using the quotient rule,

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log A divided by B is equal to log A minus log B.

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Therefore, this is going to be

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log base 4 of 256 minus log base 4 of 64.

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4 to the fourth power is 256,

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and 4 to the third power is 64,

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and 4 minus 3 is 1. That's going to be the answer.

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Try this one, log base 2 of 128 over 8.

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This is equal to log base 2 of 128 minus log base 2 of 8.

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Now, 2 to the seventh power is 128 and 2 to the third is 8.

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Seven minus 3 is 4 and that's going to be the answer.

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Here's the last example. Log base 2 of 128 times 64 divided by 8 times 16 raised to the fifth power.

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The first thing we should do is move the

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five and it's going to be distributed to everything.

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Now, every number that's on top will have a positive sign.

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The numbers on the bottom will have a negative sign.

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It's going to be log base 2 of 128 plus log base 2 of 64 since they're multiplied to each other,

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minus log base 2 of 8,

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since it's divided by,

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and also minus log base 2 of 16.

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Log base 2 of 128, we know it's 7,

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2 to the sixth power is 64,

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2 to the third is 8, 2 to the fourth is 16.

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Negative 3 and negative 4 adds up to negative 7,

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which cancels with the positive 7.

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The final answer is 5 times 6 which is 30.