Given log 2 = 0.3010 and log 3= 0.4771. Find the value of log 0.0075

Answer

To find the value of \( \log 0.0075 \), we can follow these steps:

### Step 1: Rewrite the logarithm
We start by rewriting \( 0.0075 \) in a more manageable form. We can express \( 0.0075 \) as:
\[ 0.0075 = \frac{75}{10000} = \frac{75}{10^4} \]
Thus, we have:
\[ \log 0.0075 = \log \left( \frac{75}{10^4} \right) \]

### Step 2: Apply the logarithmic property
Using the property of logarithms that states \( \log \left( \frac{m}{n} \right) = \log m - \log n \), we can separate the logarithm:
\[ \log 0.0075 = \log 75 - \log 10^4 \]

### Step 3: Simplify \( \log 10^4 \)
We know that \( \log 10^4 = 4 \log 10 \). Since \( \log 10 = 1 \), we have:
\[ \log 10^4 = 4 \]
Thus, we can rewrite our expression as:
\[ \log 0.0075 = \log 75 - 4 \]

### Step 4: Express \( \log 75 \)
Next, we can express \( 75 \) as \( 3 \times 25 \):
\[ \log 75 = \log (3 \times 25) = \log 3 + \log 25 \]
Since \( 25 = 5^2 \), we can further simplify:
\[ \log 25 = \log (5^2) = 2 \log 5 \]
So, we have:
\[ \log 75 = \log 3 + 2 \log 5 \]

### Step 5: Substitute known values
Now we substitute the known values of \( \log 2 \) and \( \log 3 \) into our equation. However, we need to find \( \log 5 \). We can use the relationship:
\[ \log 10 = \log (2 \times 5) = \log 2 + \log 5 \]
Since \( \log 10 = 1 \), we can rearrange this to find \( \log 5 \):
\[ \log 5 = 1 - \log 2 = 1 - 0.3010 = 0.6990 \]

### Step 6: Substitute \( \log 3 \) and \( \log 5 \)
Now we substitute \( \log 3 \) and \( \log 5 \) into the equation for \( \log 75 \):
\[ \log 75 = \log 3 + 2 \log 5 = 0.4771 + 2 \times 0.6990 \]
Calculating \( 2 \times 0.6990 \):
\[ 2 \times 0.6990 = 1.3980 \]
Thus:
\[ \log 75 = 0.4771 + 1.3980 = 1.8751 \]

### Step 7: Final calculation
Now we can substitute back into our equation for \( \log 0.0075 \):
\[ \log 0.0075 = \log 75 - 4 = 1.8751 - 4 = -2.1249 \]

### Conclusion
Therefore, the value of \( \log 0.0075 \) is:
\[ \log 0.0075 = -2.1249 \]

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