Today, we’re going to talk about logarithms.
The first thing you should learn is how to spell logarithms. It’s not so easy.
This would be a good word to spell on a spelling bee.
It’s logarithms.
Logarithms are really exponents.
And to illustrate, we know, for example, 4 squared equals 16.
We could also say that the logarithm – and let me just abbreviate it with log – the logarithm of 16 to the base 4 – and we’ll put that 4 down there as a subscript – of 16 of the base 4 equals 2.
So, logarithms are really exponents
Logarithms to the base 10 are called common logarithms
And let me give you some illustrations of this.
What is the logarithm of 100 to the base 10?
If the base is 10, then it’s said to be a common logarithm.
What will this be equal to? 2. Okay.
Because 10 squared is equal to 100. So, you could say the logarithm of 100 to the base 10 is equal to 2.
What about the logarithm of 1000 to the base 10?
[Student comment]
Three.
How about the logarithm of 1 to the base 10? Zero. Because 10 to the 0 is equal to 1.
One more. How about the logarithm of 0.1 to the base 10?
[Student comment]
Negative 1. Because 10 raised to the minus 1 power is 1 over 10 or 0.1
Now, just as there’s a natural exponential function, there’s also a natural logarithm.
And the natural logarithm is just logarithm to the base e.
Logarithm to the base e.
And there’s actually another notation that’s commonly used for this. Just ln. ln is understood to be logarithm to the base e.
So, from now on, let me just use ln for a natural logarithm.
This is something you’ll need to calculate. What is the natural logarithm of 100?
So, you need to have a calculator and be able to calculate natural logarithms?
What is the natural logarithm of 100?
[student comment]
4 point 60517.
That’s far enough. It actually goes further but that’s far enough.
In other words, what this is means is that e, the mathematical constant e, raised to this power, 4.60517, is equal to, approximately equal to 100.
So, I should say approximately equal to here because there’s a tiny bit of round-off here.
So, natural logarithms are used quite a bit because they’re convenient logarithms to use.
And one of the reasons they’re convenient is, again, they’re easy to differentiate
If you have an expression like this, y equals the natural logarithm of t, what is dy/dt?
Did any of you know this rule of differentiation? If y is equal to the natural log of t?
[Student comment]
Just 1 over t. So, it makes it easy to do the differentiation.
So, it’s convenient, if you have a choice of bases, it’s convenient to use natural logarithm because the differentiation is very easy
[Student comment]
Very good. If t can be negative, then it really should be the absolute value of t.
