Answer

Verified

342.3k+ views

**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 $\ln x\;{\text{or}}\;{\log _e}x$.

Recently Updated Pages

The branch of science which deals with nature and natural class 10 physics CBSE

The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths

Define absolute refractive index of a medium

Find out what do the algal bloom and redtides sign class 10 biology CBSE

Prove that the function fleft x right xn is continuous class 12 maths CBSE

Find the values of other five trigonometric functions class 10 maths CBSE

Trending doubts

How do you solve x2 11x + 28 0 using the quadratic class 10 maths CBSE

Draw a diagram of nephron and explain its structur class 11 biology CBSE

What is the color of ferrous sulphate crystals? How does this color change after heating? Name the products formed on strongly heating ferrous sulphate crystals. What type of chemical reaction occurs in this type of change.

Write the difference between soap and detergent class 10 chemistry CBSE

Give 10 examples of unisexual and bisexual flowers

Differentiate between calcination and roasting class 11 chemistry CBSE

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

What is the difference between anaerobic aerobic respiration class 10 biology CBSE

a Why did Mendel choose pea plants for his experiments class 10 biology CBSE