If `log 90=1.9542` then `log 3` equals to

Options

A. 0.9771
B. 0.6514
C. 0.4771
D. 0.3181

Answer

To find the value of \( \log 3 \) given that \( \log 90 = 1.9542 \), we can follow these steps:

### Step 1: Rewrite \( \log 90 \)
We can express \( 90 \) as the product of \( 9 \) and \( 10 \):
\[ \log 90 = \log(9 \times 10) \]

### Step 2: Apply the logarithm property
Using the property of logarithms that states \( \log(A \times B) = \log A + \log B \), we can rewrite the equation:
\[ \log 90 = \log 9 + \log 10 \]
Substituting the value of \( \log 90 \):
\[ 1.9542 = \log 9 + \log 10 \]

### Step 3: Substitute \( \log 10 \)
We know that \( \log 10 = 1 \):
\[ 1.9542 = \log 9 + 1 \]

### Step 4: Solve for \( \log 9 \)
Now, we can isolate \( \log 9 \):
\[ \log 9 = 1.9542 - 1 = 0.9542 \]

### Step 5: Rewrite \( \log 9 \)
Next, we can express \( 9 \) as \( 3^2 \):
\[ \log 9 = \log(3^2) \]

### Step 6: Apply the power property of logarithms
Using the property \( \log(A^m) = m \cdot \log A \):
\[ \log(3^2) = 2 \cdot \log 3 \]
Thus, we have:
\[ 0.9542 = 2 \cdot \log 3 \]

### Step 7: Solve for \( \log 3 \)
Now, we can isolate \( \log 3 \):
\[ \log 3 = \frac{0.9542}{2} = 0.4771 \]

### Final Answer
Therefore, the value of \( \log 3 \) is:
\[ \log 3 = 0.4771 \]

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