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When we write "log base 10" of some number x is equal to...
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...what that means is: "10 raised to what power is equal to x?"
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So, for example, if we write the log base 10 of 10,...
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...that's equal to 1, because 10, raised to the 1, is 10.
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Similarly, the log base 10 of 100 is 2, because 10, raised to the 2, is 100.
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Log base 10 of 1000 is 3, because 10 to the 3 is 1000.
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And the log base 10 of, say, 10 to the 4th, well, that's equal to 4...
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...because, well, 10 raised to the "what?" is equal to 10 to the 4th. Duh.
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We can also write log base 10 of any number, like log base 10 of 3,...
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Well, that's equal to 0.47712, and some other decimal places...
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...That's because 10 raised to 0.47712 is equal to 3....
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...You can test that on your calculator.
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We can also write logs to different bases, like log base 2.
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For example, log base 2 of 2 is 1,
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because 2 raised to the 1 is 2,
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Log base 2 of 4 is 2,..
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because 2 raised to the 2 is 4
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Log base 2 of 8 is 3, for similar reasons...
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...and so on.
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We can write log of a fractional number,
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Log base 10 of 1/10 is equal to minus 1
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because 10 to the minus 1 is, by definition, 1 tenth.
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The log base 10 of 1/100 is minus 2,..
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...because 10 to the minus 2 is 1/100
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The log base 2 of 1/4 is minus 2..,
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...because 2 to the minus 2 equals 1/4.
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The log base 2 of one eighth is -3, etc.
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In general, we use the following notation: we just write...
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...log, without any base, that refers to log base 10.
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If we write "ln", that's called the "natural logarithm", that is the log to base "e"...
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...If you don't know what "e" is, don't worry about it for now.
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When we have some other base, we write that base explicitly as a subscript of the word "log"
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Log subscript 2 is log base 2, or, in general, log subscript "a" is log base a.
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Here's some useful facts about logarithms.
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If we write the log base 10 of 10 to some number m, well that's always equal to m.
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And, in general, log base a, for some base a, whatever you want, of x to the b...
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...is always going to be, b times log base a of x.
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So you can always take the exponent of a log and bring it outside of the log.
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So that's a general fact: you can easily show that that's true.
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I won't do that here; I'll leave that as an exercise to you.
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Now, suppose that you want to compute the log to some other base besides 10...
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...like log base 2 of some number, but you only have a base 10 logarithm on your calculator...
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...typically scientific calculators only compute log to base 10,..
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...or log to base e, natural log. But you can always use this formula.
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The log base 2 of some number x is equal to the log base 10...
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of that number, divided by the log base 10 of 2.
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So you can use that on your calculator to compute log base 2 of x.
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And, more generally, for any base a the log base a of x is equal to the log base b, for anly other base b, of x...
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divided by the log base b of a.
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That's a general formula, you can also prove that that's true.
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I won't do that here, but we'll use this formula in order to calculate logs on our calculator.
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Now you can do a short quiz to test your understanding of what we just did.