Derivatives of Logarithmic Functions - Problem 3
So we've talked about the derivative of natural log. We haven't yet talked about derivatives of other logarithms. So I want to talk about that right now. First of all, recall that the derivative of natural log is 1 over x.
To get the derivatives of other logarithms, I'm going to use the change of base formula. The log base a of x equals lnx over lna. Of course you can change to any other base, but I'm going to change natural log, because I have this formula.
So if I wanted to differentiate the log of some other base a, I would first change it to this form; The derivative with respect to x of lnx over lna. Let's observe that this division by lna is just a multiplication by 1 over lna. That's a constant, so that can be pulled out. 1 over lna, times the derivative of lnx. Of course that's just 1 over x. So 1 over lna times 1 over x. That's the derivative of the log base a of x. So let's try that out on an example.
If y equals the log base 5 of x, what's the derivative? Dy/dx is the derivative of log base 5 of x. According to this formula, it's 1 over the natural log of the base, 5, times 1 over x. So 1 over ln5 times 1 over x.
A slightly harder example here. Let's find the derivative of 100 minus 3 log x. Remember, when you see log, and the base isn't written, it's assumed to be the common log, so base 10 log.
This is the derivative of 100 minus 3 log x. I can use the sum rule and constant multiple rule. I'll use both at the same time. This is the derivative of 100, minus 3 times, the derivative of log x.
Now 100, this is just a constant, Its derivative is going to be 0. I have -3 times the derivative of the log base 10 of x. That's going to be 1 over ln of 10 times 1 over x. So my answer simplifies to -3 over ln 10. That's the constant times 1 over x. That's the derivative of y equals 100 minus 3 log x.