Find the number of digit in `8^(10)` (given that `log_(10) 2 =0.3010`)

Options

A. A)19
B. B)20
C. C)17
D. D)10

Answer

To find the number of digits in \(8^{10}\), we can follow these steps:

### Step 1: Rewrite the expression
We start by rewriting \(8^{10}\) in terms of base 2:
\[ 8 = 2^3 \implies 8^{10} = (2^3)^{10} = 2^{30} \]

### Step 2: Use logarithms to find the number of digits
The number of digits \(d\) in a number \(N\) can be found using the formula:
\[ d = \lfloor \log_{10} N \rfloor + 1 \]
In our case, \(N = 8^{10} = 2^{30}\), so we need to calculate \(\log_{10}(2^{30})\):
\[ \log_{10}(2^{30}) = 30 \cdot \log_{10}(2) \]

### Step 3: Substitute the given value
We are given that \(\log_{10}(2) = 0.3010\). Now we can substitute this value into our equation:
\[ \log_{10}(2^{30}) = 30 \cdot 0.3010 = 9.03 \]

### Step 4: Calculate the number of digits
Now we can find the number of digits:
\[ d = \lfloor 9.03 \rfloor + 1 = 9 + 1 = 10 \]

### Conclusion
Thus, the number of digits in \(8^{10}\) is \(10\).

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