Question

Find the derivative of ${\log _{10}}x$.

Answer 1

Hint: We know the equation to change the base to an exponential base:
${\log _a}\left( b \right) = {\log _a}\left( e \right) \times {\log _e}\left( b \right)$
So this equation can be used to change the base 10 of the log to $e$and perform further operations.
Also$\dfrac{d}{{dx}}{\log _e}x = \dfrac{1}{x}$.
So by using the above properties and identities we can simplify and thus find the derivative of ${\log _{10}}x$.

Complete step by step solution:
Given
${\log _{10}}x........................................\left( i \right)$
We need to find the derivative of ${\log _{10}}x$. We can also see that the base of the given logarithmic term is 10, such that we cannot find its derivative directly since we only have a direct equation for the logarithmic terms with base $e$.
Now we know an equation to change the base of a logarithmic term to an exponential base:
${\log _a}\left( b \right) = {\log _a}\left( e \right) \times {\log _e}\left( b \right)..................\left( {ii}
\right)$
On comparing with (ii) we get:
$a = 10,\;b = x$
Substituting (ii) in (i) we get:
${\log _{10}}x = {\log _{10}}\left( e \right) \times {\log _e}\left( x \right)...........\left( {iii} \right)$
Now on observing (ii) we can see that the term ${\log _{10}}\left( e \right)$is a constant term. Also the next term ${\log _e}\left( x \right)$is a logarithmic term with base 10, whose derivative can be found with the standard formula:
$\dfrac{d}{{dx}}{\log _e}x = \dfrac{1}{x}$
Such that:
\[\dfrac{{d{{\log }_{10}}x}}{{dx}} = \dfrac{d}{{dx}}\left( {{{\log }_{10}}\left( e \right) \times {{\log
}_e}\left( x \right)} \right)...................\left( {iv} \right)\]
Also in (iv) the mathematical process is multiplication and thus we can take the constant term ${\log
_{10}}\left( e \right)$ as a common term outside the derivative and then perform the process of derivation.
Such that:
\[
\Rightarrow \dfrac{{d{{\log }_{10}}x}}{{dx}} = {\log _{10}}\left( e \right)\dfrac{d}{{dx}}\left( {{{\log }_e}\left( x \right)} \right) \\
\Rightarrow \dfrac{{d{{\log }_{10}}x}}{{dx}} = {\log _{10}}\left( e \right) \times
\dfrac{1}{x}......................\left( v \right) \\
\]
Therefore the derivative of ${\log _{10}}x$ is \[{\log _{10}}\left( e \right) \times \dfrac{1}{x}\]

Note:
Some of the logarithmic properties useful for solving questions of derivatives are listed below:
$
1.\;{\log _b}\left( {PQ} \right) = {\log _b}\left( P \right) + {\log _b}\left( Q \right) \\
2.\;{\log _b}\left( {\dfrac{P}{Q}} \right) = {\log _b}\left( P \right) - {\log _b}\left( Q \right) \\
3.\;{\log _b}\left( {{P^q}} \right) = q \times {\log _b}\left( P \right) \\
$
These are called Product Rule, Quotient Rule and Power Rule respectively.
Common logarithmic functions are log functions with base 10, and natural logarithmic functions are log functions with base ‘e’. Natural logarithmic functions can be represented by $\ln x\;{\text{or}}\;{\log _e}x$.