Question

The minimum value of $2{\log }_{10}\left(x\right)-{\log }_x(0.01)$, $x>1$ is

• A. $2$
• B. $4$
• C. $6$
• D. $8$

Answer 1

The correct option is B

$4$

Explanation for the correct option:

Step 1: Find the critical points.

A function $f\left(x\right)=2{\log }_{10}\left(x\right)-{\log }_x(0.01)$ is given.

According to the Logarithms change of base rule:

${\log }_b\left(a\right)=\frac{{\log }_d\left(a\right)}{{\log }_d\left(b\right)}$

Rewrite the function as follows:

$f\left(x\right)=2{\log }_{10}\left(x\right)-\frac{{\log }_{10}(0.01)}{{\log }_{10}\left(x\right)}$

Since, ${\log }_{10}(0.01)=-2$.

Therefore, $f\left(x\right)=2{\log }_{10}\left(x\right)+\frac{2}{{\log }_{10}\left(x\right)}$

Differentiate both sides with respect to $x$.

$$\begin{gathered} f\text{'}\left(x\right)=\frac{2}{x\ln 10}-\frac{2}{{\left({\log }_{10}\left(x\right)\right)}^{2}x·\ln 10} \\ \Rightarrow f\text{'}\left(x\right)=\frac{2}{x\ln 10}\left[1-\frac{1}{{\left({\log }_{10}\left(x\right)\right)}^2}\right] \end{gathered}$$ {since, $\frac{d\left[{\log }_{a}x\right]}{dx}=\frac{1}{x\ln a}$}

Substitute $f\text{'}\left(x\right)=0$ to find the critical points.

$$\begin{gathered} \frac{2}{x\ln 10}\left[1-\frac{1}{{\left({\log }_{10}\left(x\right)\right)}^2}\right]=0 \\ \Rightarrow 1-\frac{1}{{\left({\log }_{10}\left(x\right)\right)}^2}=0 \\ \Rightarrow 1=\frac{1}{{\left({\log }_{10}\left(x\right)\right)}^2} \\ \Rightarrow {\left({\log }_{10}\left(x\right)\right)}^2=1 \\ \Rightarrow {\log }_{10}\left(x\right)=\pm 1 \end{gathered}$$

Since, $a^{{\log }_a\left(b\right)}=b$.

Raise both sides to the power of $10$.

$x={10}^1,{10}^{-1}$

Since, it is given that $x>1$.

Therefore, the critical point is $x=10$.

Step 2: Find the minimum value of the given function.

Since, the critical point is $x=10$.

Compute the given function for $x=10$.

$$\begin{gathered} f\left(10\right)=2{\log }_{10}\left(10\right)-{\log }_{10}(0.01) \\ \Rightarrow f\left(10\right)=2\left(1\right)-(-2) \\ \Rightarrow f\left(10\right)=4 \end{gathered}$$

Therefore, the minimum value of the given function is $4$.

Hence, option $B$ is the correct answer.