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We are asked to solve log of x plus log of 3 is equal to

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2 log of 4 minus log of 2.

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So let's just rewrite it.

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So we have the log of x plus the log of 3

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is equal to 2 times the log of 4 minus the log of 2.

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And this is reminder: whenever you see logarithm

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written without a base, the implicit base is 10.

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So we could write 10 here, here, here and here.

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But for the rest of this example,

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I'll just skip writing 10 just cause it'll save a little bit time.

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But remember it only means log base 10.

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So this expression right over here

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is the power I have to raise 10 to get x.

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Power I have to raise 10 to get 3.

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Now that out of the way,

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let's see what logarithm properties we can use.

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So we know -- and these are all the same base --

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we know that if we have log base a of b plus log base a of c

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that this is the same thing as log base a of bc.

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And we also know --

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so let me write all of the logarithm properties that we know,

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over here.

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We also know that if we have a logarithm --

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let me write it this way actually times log base a of c

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this is equal to log base a of c to the b-th power.

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And we also know --

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and this is derived really straight from both of these, is:

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that log base a of b minus log base a of c

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that this is equal to log base a of b over c.

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And this is really straight derived from these two right over here.

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Now with that out of the way let's see what we can apply.

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So right over here we have -- all of the logs are the same base --

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we have logarithm of x plus logarithm of 3,

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so by this property right over here --

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the sum of the logarithms of the same base

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this is going to be equal to log base 10 of 3 times x of 3x.

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Then based on this property right over here

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this thing can be rewritten, this is going to be equal to

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this can be written as log base 10 of 4 to the second power,

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which is really just 16.

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and then we still have minus logarithm base 10 of 2.

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And now using this last property --

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we know we have one logarithm minus another logarithm.

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This is going to be equal to log base 10 of 16 over 2,

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16 divided by 2, which is the same thing as 8.

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So right-hand side simplifies to log base 10 of 8

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the left-hand side is log base 10 of 3x.

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So if 10 to some power is going to be equal 3x,

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10 to the same power is going to be equal to 8.

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So 3x must be equal to 8.

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3x is equal to 8.

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And then we can divide both sides by 3.

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You get x is equal to 8 over 3.

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One way this little step here I said: look this is an exponent.

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If I raise 10 to this exponent I get 3x,

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10 to this exponent I get 8.

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So 8 and 3x must be the same thing.

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One other way you could've thought about is:

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let's take 10 to this power on both sides.

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So you could say 10 to this power and then 10 to this power over here.

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If I raise 10 to the power that I need to raise 10 to get 3x

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well I'm just going to get 3x!

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If I raise 10 to the power that I need to raise 10 to get 8

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I'm just going to get 8.

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So once again you get 3x is equal to 8.

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And you can simplify,you get x is equal to 8/3.