If log2=0.301 and log3=0.477, find the value of log(3.375).

Answer

To find the value of \( \log(3.375) \) given \( \log(2) = 0.301 \) and \( \log(3) = 0.477 \), we can follow these steps:

### Step 1: Express \( 3.375 \) in a simpler form
First, we can express \( 3.375 \) as a fraction:
\[ 3.375 = \frac{81}{24} \]
This is because \( 3.375 = \frac{27}{8} = \frac{3^3}{2^3} = \frac{3^3}{2^3} = \frac{3^3}{2^3} = \frac{81}{24} \).

### Step 2: Apply the logarithm property
Using the property of logarithms that states \( \log\left(\frac{a}{b}\right) = \log(a) - \log(b) \), we can rewrite:
\[ \log(3.375) = \log\left(\frac{81}{24}\right) = \log(81) - \log(24) \]

### Step 3: Simplify \( \log(81) \) and \( \log(24) \)
Next, we can express \( 81 \) and \( 24 \) in terms of their prime factors:
\[ 81 = 3^4 \quad \text{and} \quad 24 = 2^3 \times 3^1 \]
Now we can write:
\[ \log(81) = \log(3^4) = 4 \log(3) \]
\[ \log(24) = \log(2^3 \times 3^1) = \log(2^3) + \log(3) = 3 \log(2) + \log(3) \]

### Step 4: Substitute the logarithm values
Now substituting these into our equation:
\[ \log(3.375) = 4 \log(3) - (3 \log(2) + \log(3)) \]
This simplifies to:
\[ \log(3.375) = 4 \log(3) - 3 \log(2) - \log(3) = (4 - 1) \log(3) - 3 \log(2) = 3 \log(3) - 3 \log(2) \]
Factoring out the 3 gives:
\[ \log(3.375) = 3(\log(3) - \log(2)) \]

### Step 5: Substitute the known values
Now we can substitute the known values of \( \log(2) \) and \( \log(3) \):
\[ \log(3.375) = 3(0.477 - 0.301) \]
Calculating inside the parentheses:
\[ 0.477 - 0.301 = 0.176 \]
Now substituting back:
\[ \log(3.375) = 3 \times 0.176 = 0.528 \]

### Final Answer
Thus, the value of \( \log(3.375) \) is:
\[ \boxed{0.528} \]