Given `log 2 = 0.3010` and log 3= 0.4771, find the value of log 45.

Answer

To find the value of \( \log 45 \) given that \( \log 2 = 0.3010 \) and \( \log 3 = 0.4771 \), we can follow these steps:

### Step 1: Express \( \log 45 \) in terms of known logarithms
We can start by expressing \( 45 \) as a product of its prime factors:
\[ 45 = 9 \times 5 \]
Thus, we can write:
\[ \log 45 = \log(9 \times 5) \]

### Step 2: Use the logarithmic property
Using the property of logarithms that states \( \log(m \times n) = \log m + \log n \), we can rewrite the equation:
\[ \log 45 = \log 9 + \log 5 \]

### Step 3: Express \( \log 9 \) in terms of \( \log 3 \)
Since \( 9 = 3^2 \), we can use the power rule of logarithms, which states \( \log(n^x) = x \cdot \log n \):
\[ \log 9 = \log(3^2) = 2 \cdot \log 3 \]
Substituting this back into our equation gives:
\[ \log 45 = 2 \cdot \log 3 + \log 5 \]

### Step 4: Express \( \log 5 \) using \( \log 10 \) and \( \log 2 \)
We know that \( 10 = 2 \times 5 \), so we can express \( \log 5 \) as:
\[ \log 5 = \log 10 - \log 2 \]
Given that \( \log 10 = 1 \), we can substitute:
\[ \log 5 = 1 - \log 2 \]

### Step 5: Substitute known values
Now, substituting \( \log 3 \) and \( \log 2 \) into our equation:
\[ \log 45 = 2 \cdot \log 3 + (1 - \log 2) \]
Using the values \( \log 2 = 0.3010 \) and \( \log 3 = 0.4771 \):
\[ \log 45 = 2 \cdot 0.4771 + (1 - 0.3010) \]

### Step 6: Calculate the values
Calculating \( 2 \cdot 0.4771 \):
\[ 2 \cdot 0.4771 = 0.9542 \]
Now substituting this into the equation:
\[ \log 45 = 0.9542 + (1 - 0.3010) \]
Calculating \( 1 - 0.3010 \):
\[ 1 - 0.3010 = 0.6990 \]
Now, adding these two results together:
\[ \log 45 = 0.9542 + 0.6990 = 1.6532 \]

### Final Answer
Thus, the value of \( \log 45 \) is:
\[ \log 45 = 1.6532 \]