The least value of expression `2 log_(10)x - log_(x) (1//100)` for `x gt 1` is:

Options

A. 2
B. 3
C. 4
D. 5

Answer

To find the least value of the expression \( 2 \log_{10} x - \log_{x} \left( \frac{1}{100} \right) \) for \( x > 1 \), we will simplify and analyze the expression step by step.

### Step 1: Rewrite the expression
Start with the expression:
\[ 2 \log_{10} x - \log_{x} \left( \frac{1}{100} \right) \]

### Step 2: Simplify the logarithm
Recall that \( \frac{1}{100} = 10^{-2} \). Therefore, we can rewrite the logarithm:
\[ \log_{x} \left( \frac{1}{100} \right) = \log_{x} \left( 10^{-2} \right) = -2 \log_{x} (10) \]

### Step 3: Change of base formula
Using the change of base formula, we have:
\[ \log_{x} (10) = \frac{\log_{10} (10)}{\log_{10} (x)} = \frac{1}{\log_{10} (x)} \]
Thus,
\[ \log_{x} \left( \frac{1}{100} \right) = -2 \cdot \frac{1}{\log_{10} (x)} = -\frac{2}{\log_{10} (x)} \]

### Step 4: Substitute back into the expression
Now substitute this back into the original expression:
\[ 2 \log_{10} x - \left(-\frac{2}{\log_{10} x}\right) = 2 \log_{10} x + \frac{2}{\log_{10} x} \]

### Step 5: Let \( t = \log_{10} x \)
Let \( t = \log_{10} x \). Since \( x > 1 \), we have \( t > 0 \). The expression becomes:
\[ 2t + \frac{2}{t} \]

### Step 6: Find the minimum value
To find the minimum value of \( 2t + \frac{2}{t} \), we can use the AM-GM inequality:
\[ \frac{2t + \frac{2}{t}}{2} \geq \sqrt{2t \cdot \frac{2}{t}} = 2 \]
Thus,
\[ 2t + \frac{2}{t} \geq 4 \]

### Step 7: Determine when equality holds
Equality in the AM-GM inequality holds when \( 2t = \frac{2}{t} \), or \( t^2 = 1 \). Therefore, \( t = 1 \) (since \( t > 0 \)).

### Step 8: Calculate the minimum value
Substituting \( t = 1 \) back into the expression:
\[ 2(1) + \frac{2}{1} = 2 + 2 = 4 \]

### Conclusion
Thus, the least value of the expression \( 2 \log_{10} x - \log_{x} \left( \frac{1}{100} \right) \) for \( x > 1 \) is:
\[ \boxed{4} \]