1
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we're asked to simply log base 3 of 27x

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and frankly this is already quite simple,

3
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but I'm assuming they want us to use

4
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some logarithm properties and manipulate this some way

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maybe actually make it a little more complicated

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and let's give this our best shot at it

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so the logarithm property that jumps out at me

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because this, right over here, is saying

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what power do we have to raise 3 to in order to get 27x

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27x is the same thing as 27 times x

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and so the logarithm property it seems like they want us to use is

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log base b of a times c

13
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this is equal to

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the logarithm base b of a plus logarithm base b of c

15
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now this comes straight out of the exponent properties

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that if you have two exponents with the same base

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you can add the exponents

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now let me make this a little bit clearer to you

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now if this part is a little confusing the important part for this example is

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is that you know how to apply this,

21
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but it's even better if you know the intuition

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so let's say that log, let's say that log base b of a times c is equal to x

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so this thing right over here evaluates to x

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let's say that this thing right over here evaluates to y

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so log base b of a is equal to y

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and let's say that that this thing over here evaluates to z

27
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so log base b of c is equal to z

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now, what we know is

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this thing right over here, this thing right over here

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or this thing right over here tells us

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tells us that b, to the x power is equal to a times c

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now, this right over here is telling us that

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b to the y power is equal to a

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and this over here is telling us

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that b to the z power is equal to c

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let me do that in that same green

37
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so i'm just writing the same truth

38
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i'm writing as an exponential function

39
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or exponential equation

40
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instead of a logarithmic equation

41
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so b to the zth power is equal to c

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these are all, this is the same statement

43
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this is the same statement

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or they're the same truth said in a different way

45
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and this is the same truth said in a different way

46
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well if we know that, if we know that

47
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a is equal to this, is equal to b to the y,

48
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we can, and c is equal to b-z

49
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then we can write

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b to the x power is equal to b to the y power

51
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that's what a is, we know that already

52
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times b to the z power

53
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times b to the z power

54
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and we know, from our exponent properites

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we know from our exponent properties

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That if we take b to the y times b to the z

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this is the same thing as:

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b to the, I'll do it in neutral color, b to the y plus z power

59
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this came straight out of our exponent properties.

60
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And so if b to the y plus z power is the same thing

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b to the x power -- that tells us that x must be equal to y plus z.

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x must be equal to y plus z.

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If this is confusing to you -- don't worry too much.

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The important thing, or at least the first important thing is

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that you know how to apply it and then you can think about this

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a little bit more and you can even try it out with some numbers.

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You have to just realise that logarithms are really just exponents.

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I know, when people at first tell me: what does that mean?

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But when you evaluate a logarithm - you're getting an exponent

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that you would have to raise b to, in order to get a times c.

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But let's just apply this property right over here.

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So if we apply to this one we know that

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log base 3 of 27 times x -- I'll write it that way --

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is equal to log base 3 of 27 plus log base 3 of x.

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And then this right over here we can evaluate,

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this tells us: what power do I have to raise 3 to get to 27.

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You can view it this way: 3 to the question mark is equal to 27.

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Well, 3 to the 3rd power is equal to 27.

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3 times 3 is 9, times 3 is 27.

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To this right over here evaluates to 3.

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So if we were to simplify -- or I guess I wouldn't even call it

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simplifying, I'll just call it expanding it out or using this property.

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We now have two terms, where when we started of with one term.

84
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Actually if we started with this, I'd say that this is the more simple version of it.

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But when we rewrite it, this first term becomes 3.

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So this first term becomes 3

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and then we're left with plus log base 3 of x.

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So this is just an alternate way of writing this original statement.

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Log base 3 of 27x.

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So once again -- not clear that this is simpler than this right over here.

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It's just another way of writing it using logarithm properties.