Question

The differential coefficient of log₁₀ x with respect to log_x 10 is
1. \(\frac{{{x^2}}}{{100}}\)
2. (log_x 10)²
3. -(log₁₀ x)²
4. 1

Answer 1

Correct Answer - Option 3 : -(log₁₀ x)²

Concept:

Property of logarithm:  \({\log _x}10 = \frac{1}{{{{\log }_{10}}x}}\)

Calculation:

Let y = log10 x and z =  logx 10

Now, yz = (log10 x) × (logx 10)

⇒ yz = 1

Differentiating with respect to z, we get

\(\rm\\y\left( \frac {dz}{dz} \right) + z\left( {\frac{{dy}}{{dz}}} \right) = 0\\ y + z\left( {\frac{{dy}}{{dz}}} \right) = 0\\\frac{{dy}}{{dz}} = - \frac{y}{z}\\ = - \frac{{{{\log }_{10}}x}}{{{{\log }_x}10}}\\ = - \frac{{{{\log }_{10}}x}}{{\frac{1}{{{{\log }_{10}}x}}}}\\ = - {\left( {{{\log }_{10}}x} \right)^2}\)