Find number of digits in `875^(16)`, given log2=.3010300,log7=.8450980.

Answer

To find the number of digits in \( 875^{16} \), we can use the formula for the number of digits of a number \( n \), which is given by:

\[ \text{Number of digits} = \lfloor \log_{10} n \rfloor + 1 \]

In this case, \( n = 875^{16} \). Therefore, we need to calculate \( \log_{10} (875^{16}) \).

### Step 1: Use the logarithmic property
Using the property of logarithms, we can express this as:

\[ \log_{10} (875^{16}) = 16 \cdot \log_{10} (875) \]

### Step 2: Break down \( 875 \)
Next, we can express \( 875 \) in terms of its prime factors:

\[ 875 = 7 \times 125 = 7 \times (5^3) \]

### Step 3: Apply the logarithmic property
Now, we can write:

\[ \log_{10} (875) = \log_{10} (7 \times 5^3) = \log_{10} (7) + \log_{10} (5^3) \]

Using the property of logarithms \( \log (a^b) = b \cdot \log (a) \):

\[ \log_{10} (5^3) = 3 \cdot \log_{10} (5) \]

Thus, we have:

\[ \log_{10} (875) = \log_{10} (7) + 3 \cdot \log_{10} (5) \]

### Step 4: Express \( \log_{10} (5) \) in terms of \( \log_{10} (2) \)
Since \( \log_{10} (5) = \log_{10} \left( \frac{10}{2} \right) = \log_{10} (10) - \log_{10} (2) = 1 - \log_{10} (2) \), we can substitute this into our equation:

\[ \log_{10} (875) = \log_{10} (7) + 3 \cdot (1 - \log_{10} (2)) \]

### Step 5: Substitute known logarithm values
We are given:

- \( \log_{10} (2) = 0.3010300 \)
- \( \log_{10} (7) = 0.8450980 \)

Substituting these values gives:

\[ \log_{10} (875) = 0.8450980 + 3 \cdot (1 - 0.3010300) \]

Calculating \( 1 - 0.3010300 \):

\[ 1 - 0.3010300 = 0.6989700 \]

Now substituting back:

\[ \log_{10} (875) = 0.8450980 + 3 \cdot 0.6989700 \]

Calculating \( 3 \cdot 0.6989700 \):

\[ 3 \cdot 0.6989700 = 2.0969100 \]

Thus:

\[ \log_{10} (875) = 0.8450980 + 2.0969100 = 2.9420080 \]

### Step 6: Calculate \( \log_{10} (875^{16}) \)
Now we can find:

\[ \log_{10} (875^{16}) = 16 \cdot 2.9420080 = 47.0721280 \]

### Step 7: Find the number of digits
Finally, we apply the formula for the number of digits:

\[ \text{Number of digits} = \lfloor 47.0721280 \rfloor + 1 = 47 + 1 = 48 \]

Thus, the number of digits in \( 875^{16} \) is **48**.

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