If log 4= 0.602 and log 27 = 1.431 , find :
` log 8 `

Answer

To find the value of \( \log 8 \) using the given values of \( \log 4 \) and \( \log 27 \), we can follow these steps:

### Step 1: Express \( \log 8 \) in terms of \( \log 2 \)
We know that \( 8 \) can be expressed as \( 2^3 \). Therefore, we can use the logarithmic identity:
\[ \log 8 = \log(2^3) = 3 \log 2 \]

### Step 2: Find \( \log 2 \) using \( \log 4 \)
We know that \( 4 \) can be expressed as \( 2^2 \). Thus, we can write:
\[ \log 4 = \log(2^2) = 2 \log 2 \]
Given that \( \log 4 = 0.602 \), we can set up the equation:
\[ 2 \log 2 = 0.602 \]

### Step 3: Solve for \( \log 2 \)
To find \( \log 2 \), we can divide both sides of the equation by 2:
\[ \log 2 = \frac{0.602}{2} = 0.301 \]

### Step 4: Substitute \( \log 2 \) back into the expression for \( \log 8 \)
Now that we have \( \log 2 \), we can substitute it back into our expression for \( \log 8 \):
\[ \log 8 = 3 \log 2 = 3 \times 0.301 \]

### Step 5: Calculate \( \log 8 \)
Now we can perform the multiplication:
\[ \log 8 = 3 \times 0.301 = 0.903 \]

### Final Answer
Thus, the value of \( \log 8 \) is:
\[ \log 8 = 0.903 \]

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