Log base 10 not equal to log while differentiating?

Question

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!

Answer 1

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 $$\frac{d}{dx} \log_{10} x = \frac{d}{dx}\left(\frac{\ln x}{\ln 10}\right) = \frac{1}{x \log 10}.$$

Answer 2

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.