Question

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

Answer 1

Hint: 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.