Question

What is the derivative of $y={{\log }_{10}}\left( x \right)?$

Answer 1

To solve this question, use the basic concepts of logarithms and basic differentiation formulae. We convert the given function in the question which is of a base 10 to the base e. This can be done by using the formula ${{\log }_{a}}b=\dfrac{{{\log }_{e}}b}{{{\log }_{e}}a}.$ By doing this, we end up with $\ln$ terms and we know the derivative of $\ln x$ is $\dfrac{1}{x}.$ Using these formulae and simplifying the above equation, we obtain the result.

Complete step by step solution:
The function given to us is $y={{\log }_{10}}\left( x \right).$ The given question requires us to find the derivative of ${{\log }_{10}}x.$ We know the basic differentiation formula for $\ln x$ is given as:
$\Rightarrow \dfrac{d}{dx}\left( \ln x \right)=\dfrac{1}{x}\ldots \ldots \left( 1 \right)$
This formula is applicable only if the base of the $\log$ function is exponent or $e.$ Now to convert the given question in terms of $\ln ,$ we use the formula given by:
$\Rightarrow {{\log }_{a}}b=\dfrac{{{\log }_{e}}b}{{{\log }_{e}}a}$
Here a represents the base of the logarithmic function. The given question has a base 10. Substituting this in the above equation,
$\Rightarrow {{\log }_{10}}x=\dfrac{{{\log }_{e}}x}{{{\log }_{e}}10}$
We know that ${{\log }_{e}}a=\ln a.$ Therefore, replacing the $\log$ terms by $\ln ,$
$\Rightarrow {{\log }_{10}}x=\dfrac{\ln x}{\ln 10}$
It is known that $\ln 10$ is a constant. Therefore, applying differentiation to both sides of the above equation,
$\Rightarrow \dfrac{d}{dx}\left( {{\log }_{10}}x \right)=\dfrac{1}{\ln 10}.\dfrac{d}{dx}\left( \ln x \right)$
Using the formula given in equation 1,
$\Rightarrow \dfrac{d}{dx}\left( {{\log }_{10}}x \right)=\dfrac{1}{\ln 10}.\dfrac{1}{x}$
We know the value of $\ln 10=2.3026.$ Substituting this in the above equation,
$\Rightarrow \dfrac{d}{dx}\left( {{\log }_{10}}x \right)=\dfrac{1}{2.3026.x}$

Hence, the derivative of $y={{\log }_{10}}\left( x \right)$ is $\dfrac{1}{x.\ln 10}$ or $\dfrac{1}{2.3026.x}.$

Note: It is essential to know the basic formulae of conversion from $\log$ function to $\ln$ function. It is important to note that $\log$ is used when the base can have any value but $\ln$ is used only for base e. Students are required to know the standard differentiation formulae to solve such questions with ease.