The least value of 2`log_100 a-log_a 0.0001 , a >1`

Options

A. 2
B. 3
C. 4
D. 6

Answer

To find the least value of the expression \(2 \log_{100} a - \log_a 0.0001\) for \(a > 1\), we will proceed step by step.

### Step 1: Rewrite the logarithmic expression
First, we rewrite the expression using the properties of logarithms. We know that:
\[ \log_a 0.0001 = \log_a (10^{-4}) = -4 \log_a 10 \]
Thus, we can rewrite the original expression as:
\[ 2 \log_{100} a - (-4 \log_a 10) = 2 \log_{100} a + 4 \log_a 10 \]

### Step 2: Change the base of the logarithm
Next, we can change the base of the logarithm from 100 to 10. We use the change of base formula:
\[ \log_{100} a = \frac{\log_{10} a}{\log_{10} 100} = \frac{\log_{10} a}{2} \]
So, we can substitute this back into our expression:
\[ 2 \left(\frac{\log_{10} a}{2}\right) + 4 \log_a 10 = \log_{10} a + 4 \log_a 10 \]

### Step 3: Rewrite \(\log_a 10\)
Using the change of base formula again, we can express \(\log_a 10\) as:
\[ \log_a 10 = \frac{1}{\log_{10} a} \]
Substituting this into the expression gives:
\[ \log_{10} a + 4 \left(\frac{1}{\log_{10} a}\right) \]

### Step 4: Let \(x = \log_{10} a\)
Let \(x = \log_{10} a\). Since \(a > 1\), we have \(x > 0\). The expression now becomes:
\[ f(x) = x + \frac{4}{x} \]

### Step 5: Find the minimum value of \(f(x)\)
To find the minimum value of \(f(x)\), we can take the derivative and set it to zero:
\[ f'(x) = 1 - \frac{4}{x^2} \]
Setting the derivative equal to zero:
\[ 1 - \frac{4}{x^2} = 0 \implies \frac{4}{x^2} = 1 \implies x^2 = 4 \implies x = 2 \quad (\text{since } x > 0) \]

### Step 6: Evaluate \(f(x)\) at \(x = 2\)
Now, we evaluate \(f(x)\) at \(x = 2\):
\[ f(2) = 2 + \frac{4}{2} = 2 + 2 = 4 \]

### Step 7: Verify that this is a minimum
To confirm that this is indeed a minimum, we can check the second derivative:
\[ f''(x) = \frac{8}{x^3} \]
Since \(f''(x) > 0\) for \(x > 0\), \(f(x)\) is concave up, confirming that \(x = 2\) is a minimum.

### Conclusion
Thus, the least value of the expression \(2 \log_{100} a - \log_a 0.0001\) for \(a > 1\) is:
\[ \boxed{4} \]