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I was looking at sample questions from my textbook and I came across something interesting that I need a little help understanding

The question was to find the derivative of:

$\log_{10} (\frac{7}{\sqrt {x+3}})$

My solution was:

$\frac{1}{2(x+3)}$

My problem is that when I later tried to check my answer online, it resulted in:

$\frac {1}{2(x+3)*\log(10)}$

So this led me to believe that, in the case of deriving with logs, $\log_{10}$ is not the same thing as $\log$, yet in every other scenario it is the same thing.

My question is, are these two cases actually different, is one of the results wrong, or are both results the same with the second one being a simple expansion (which I don't think it is but I could be wrong).

Any help will be great, this is a really interesting case. Thank you!

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Often, at calculus and beyond, $\log$ refers to the natural logarithm $\ln$, which we might also write for emphasis as $\log_e$, and we denote the perhaps more familiar base-$10$ logarithm as $\log_{10}$.

The derivative of $\ln$ is $$\frac{d}{dx} \ln x = \frac{1}{x}$$ (in fact, this can be taken as part of the definition of $\ln$), and the base conversion identity for logarithms gives $$\frac{d}{dx} \log_{10} x = \frac{d}{dx}\left(\frac{\ln x}{\ln 10}\right) = \frac{1}{x \log 10}.$$

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  • $\begingroup$ Great answer. Not trying to bother, but I'm just wondering what the point of this is. Since both ln and log would be equal at this point, why not just keep only ln in use and set the assumed base to "e," and set it to a different base if needed? $\endgroup$ – mrybak834 Oct 14 '14 at 10:02
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    $\begingroup$ Thank you, and no bother: The symbol $\ln$ is an abbreviation for the Latin logarithmus naturalis, so it would be an inappropriate name for a logarithm with a general base. In practice, mathematicians often do just the opposite of what you suggest, namely not use $\ln$, use $\log$ for the natural logarithm, and use $\log_b$ for the base-$b$ logarithm. $\endgroup$ – Travis Willse Oct 14 '14 at 10:14
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Assume your function was $\dfrac{d}{dx}\log_{10}(f(x)) = \dfrac{d}{dx} \dfrac{\ln(f(x))}{\ln 10} = \dfrac{f'(x)}{f(x)\cdot \ln10}$.

This is because $\log_{10}(a) = \dfrac{\ln a}{\ln 10}$, where $\ln$ is natural logarithm. Also $\dfrac{d}{dx}\ln(f(x)) = \dfrac{f'(x)}{f(x)}$ holds only when base is $e$, i.e. $\log$ is $\ln$ i.e. natural logarithm.

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