Quotient Rule Involving A Logarithm

**Bashyboy** · May 5th 2012, 02:42 PM

The problem is: $g(x) = \frac{10\log_{4} x}{x}$

By doing the quotient rule I get: $u = 10\log_{4} x$ and $u' = \frac{10}{ln 4\times{x}}$ and $v = x$ and $v' = 1$

When I apply the quotient rule I get: g'(x) = (10 - ln 4 log_4 x)/(x^2 ln 4) (sorry, I couldn't figure out the latex for this part.)

This answer is not even remotely close to the actual answer in the book. What did I do wrong?

**skeeter** · May 5th 2012, 02:56 PM

Originally Posted by Bashyboy

The problem is: $g(x) = \frac{10\log_{4} x}{x}$

By doing the quotient rule I get: $u = 10\log_{4} x$ and $u' = \frac{10}{ln 4\times{x}}$ and $v = x$ and $v' = 1$

When I apply the quotient rule I get: g'(x) = (10 - ln 4 log_4 x)/(x^2 ln 4) (sorry, I couldn't figure out the latex for this part.)

This answer is not even remotely close to the actual answer in the book. What did I do wrong?

note the change of base formula ...

$\log_b{a} = \frac{\ln{a}}{\ln{b}}$

$g(x) = \frac{10\log_{4} x}{x} = \frac{10}{\ln{4}} \cdot \frac{\ln{x}}{x}$

$g'(x) = \frac{10}{\ln{4}} \cdot \frac{1 - \ln{x}}{x^2}$

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