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Derivatives of Logarithmic Functions - Problem 3
PhD. in Mathematics
Norm was 4th at the 2004 USA Weightlifting Nationals! He still trains and competes occasionally, despite his busy schedule.
To find the derivative of other logarithmic functions, you must use the change of base formula: loga(x)= ln(x)/ln(a). With this, you can derive logarithmic functions with any base. For example, if f(x)=log3(x), then f(x)=ln(x)/ln(3). Since ln(3) is a constant, you can derive this as you would any other natural log function; f'(x)=(1/ln(3))*1/x=1/(ln(3)*x). Remember that if no base is specified, the log(x) has base 10.
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.
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