When we write "log base 10" of some number x is equal to...

...what that means is: "10 raised to what power is equal to x?"

So, for example, if we write the log base 10 of 10,...

...that's equal to 1, because 10, raised to the 1, is 10.

Similarly, the log base 10 of 100 is 2, because 10, raised to the 2, is 100.

Log base 10 of 1000 is 3, because 10 to the 3 is 1000.

And the log base 10 of, say, 10 to the 4th, well, that's equal to 4...

...because, well, 10 raised to the "what?" is equal to 10 to the 4th. Duh.

We can also write log base 10 of any number, like log base 10 of 3,...

Well, that's equal to 0.47712, and some other decimal places...

...That's because 10 raised to 0.47712 is equal to 3....

...You can test that on your calculator.

We can also write logs to different bases, like log base 2.

For example, log base 2 of 2 is 1,

because 2 raised to the 1 is 2,

Log base 2 of 4 is 2,..

because 2 raised to the 2 is 4

Log base 2 of 8 is 3, for similar reasons...

...and so on.

We can write log of a fractional number,

Log base 10 of 1/10 is equal to minus 1

because 10 to the minus 1 is, by definition, 1 tenth.

The log base 10 of 1/100 is minus 2,..

...because 10 to the minus 2 is 1/100

The log base 2 of 1/4 is minus 2..,

...because 2 to the minus 2 equals 1/4.

The log base 2 of one eighth is -3, etc.

In general, we use the following notation: we just write...

...log, without any base, that refers to log base 10.

If we write "ln", that's called the "natural logarithm", that is the log to base "e"...

...If you don't know what "e" is, don't worry about it for now.

When we have some other base, we write that base explicitly as a subscript of the word "log"

Log subscript 2 is log base 2, or, in general, log subscript "a" is log base a.

Here's some useful facts about logarithms.

If we write the log base 10 of 10 to some number m, well that's always equal to m.

And, in general, log base a, for some base a, whatever you want, of x to the b...

...is always going to be, b times log base a of x.

So you can always take the exponent of a log and bring it outside of the log.

So that's a general fact: you can easily show that that's true.

I won't do that here; I'll leave that as an exercise to you.

Now, suppose that you want to compute the log to some other base besides 10...

...like log base 2 of some number, but you only have a base 10 logarithm on your calculator...

...typically scientific calculators only compute log to base 10,..

...or log to base e, natural log. But you can always use this formula.

The log base 2 of some number x is equal to the log base 10...

of that number, divided by the log base 10 of 2.

So you can use that on your calculator to compute log base 2 of x.

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...

divided by the log base b of a.

That's a general formula, you can also prove that that's true.

I won't do that here, but we'll use this formula in order to calculate logs on our calculator.

Now you can do a short quiz to test your understanding of what we just did.