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One problem I've been asked to solve is giving me some trouble on the particular sequence to solve. Find the differential of

$$g(t)=\frac{10 \log_{4}t}{t}$$

Looking at the problem, you can see that the 10 is a constant, and can be pulled out of the function and that the quotient rule applies

$$g'(t)= 10[\frac{\log_{4}t}{t}]$$ $$g'(t)=10[\frac{\log_{4}t[\frac{d}{dx}(t)]-t[\frac{d}{dx}\log_4t]}{t^2}]$$ $$g'(t)= 10[\frac{\log_4t-t(\frac{1}{\ln(4)t})}{t^2}]$$ $$g'(t)= 10[\frac{\log_4t-\frac{t}{\ln(4)t}}{t^2}]$$ $$g'(t)=10[\frac{t^2}{\log_4- 2t \ln(4)}]$$

I got stuck right there, since I'm not sure my last line works, so I took a look at the book solution, which is shown below:

$$g(t)= \frac{10 \log_4t}{t} = \frac{10}{\ln 4}\cdot\frac{\ln(t)}{t}$$

So my questions are:

$\cdot$There are 3 elements of the numerator, so how does the $\log_4$ morph into the denominator of $\frac{10}{\ln 4}$? $\cdot$The book example shows that the quotient differentiation is done only on $\frac{\ln(t)}{t}$, but why not on the other fraction? Is it due to those values being constants and not open to differentiation?
$\cdot$Is the last line of my own work accurate?

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Let $z=\log_4 t$. Then $4^z=t$. Take the $\ln$ of both sides. We get $z\ln 4=\ln t$. So $\log_4 t=\ln t/\ln 4$. (This may be also found from a formula you once memorized.) So you want to differentiate $10\ln t/(t\ln 4)$. This should not give you trouble. –  André Nicolas Jul 17 '11 at 21:48
    
About your various questions, there are some errors, including algebra errors. You quoted the derivative of $\log_4 t$ correctly, presumably from a list of formulas. I would not be able to remember such a thing, so I would always derive it as in the comment above. –  André Nicolas Jul 17 '11 at 22:22
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1 Answer

up vote 3 down vote accepted

The expression $\log_4(t)$ is not multiplication. It is the application of a function. Your mystery will be revealed by the change of base formula, $$\log_4(t) = {\log(t)\over \log(4)}.$$

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