If `log_(10)2` = 0.3010, then find the number of digits in `(64)^(10)`.

Answer

To find the number of digits in \( (64)^{10} \), we can follow these steps:

### Step 1: Express the number in logarithmic form
Let \( y = (64)^{10} \). To find the number of digits in \( y \), we can use the formula:
\[ \text{Number of digits} = \lfloor \log_{10} y \rfloor + 1 \]

### Step 2: Apply logarithm to \( y \)
Using the property of logarithms, we can rewrite \( \log_{10} y \):
\[ \log_{10} y = \log_{10}((64)^{10}) = 10 \cdot \log_{10}(64) \]

### Step 3: Express \( 64 \) as a power of \( 2 \)
Since \( 64 = 2^6 \), we can substitute this into our logarithmic expression:
\[ \log_{10}(64) = \log_{10}(2^6) = 6 \cdot \log_{10}(2) \]

### Step 4: Substitute the value of \( \log_{10}(2) \)
Given that \( \log_{10}(2) = 0.3010 \), we can substitute this value:
\[ \log_{10}(64) = 6 \cdot 0.3010 = 1.806 \]

### Step 5: Calculate \( \log_{10} y \)
Now we can find \( \log_{10} y \):
\[ \log_{10} y = 10 \cdot \log_{10}(64) = 10 \cdot 1.806 = 18.06 \]

### Step 6: Find the number of digits
Using the formula for the number of digits:
\[ \text{Number of digits} = \lfloor 18.06 \rfloor + 1 = 18 + 1 = 19 \]

Thus, the number of digits in \( (64)^{10} \) is **19**.

---