Question

What is the derivative of ${\log _3}x$?

Answer 1

Derivative of a function gives the rate of change of the function value with respect to change in its argument value. To find the derivative of ${\log _3}x$, we will first apply the base change rule of logarithm and change the base of ${\log _3}x$ from $3{\text{ to e}}$. Then we will apply the formula for the derivative of $\ln x$ and find the derivative of ${\log _3}x$.

Complete step by step answer:
Let,
$y = {\log _3}x$
Now we know from the base change property of logarithm that,
${\log _3}x = \dfrac{{{{\log }_e}x}}{{{{\log }_e}3}}$
So, using this we get;
$\Rightarrow y = \dfrac{{{{\log }_e}x}}{{{{\log }_e}3}}$
Now differentiating both sides we get;
$\Rightarrow \dfrac{{dy}}{{dx}} = \dfrac{d}{{dx}}\left( {\dfrac{{{{\log }_e}x}}{{{{\log }_e}3}}} \right)$
Now we know ${\log _e}3$ is a constant. So, we will take it out of the differentiation sign.
$\Rightarrow \dfrac{{dy}}{{dx}} = \dfrac{1}{{{{\log }_e}3}} \times \dfrac{d}{{dx}}\left( {{{\log }_e}x} \right)$
Now we know that, $\dfrac{{d\ln x}}{{dx}} = \dfrac{1}{x}$, so we get;
$\Rightarrow \dfrac{{dy}}{{dx}} = \dfrac{1}{{x{{\log }_e}3}}$
We can also write it as;
$\Rightarrow \dfrac{{dy}}{{dx}} = \dfrac{1}{{x\ln 3}}$

Note:
One mistake that most of the students commit in these types of questions is that they simply apply the formula that $\dfrac{d}{{dx}}\log x = \dfrac{1}{x}$. This is because this formula is valid only when the base is $e$. But in the question the base of logarithm is $3$. So, we have to change the base first and then do the differentiation.