If `log_(10)3=0*477` , the number of digits in `3^(40)` is

Options

A. `18`
B. `19`
C. `20`
D. `21`

Answer

To find the number of digits in \(3^{40}\), we can use the formula for the number of digits \(d\) in a number \(N\), which is given by:

\[ d = \lfloor \log_{10} N \rfloor + 1 \]

In this case, \(N = 3^{40}\). Therefore, we need to calculate:

\[ d = \lfloor \log_{10} (3^{40}) \rfloor + 1 \]

### Step 1: Use the property of logarithms
We can use the property of logarithms that states \(\log_{10} (a^b) = b \cdot \log_{10} a\). Thus, we have:

\[ \log_{10} (3^{40}) = 40 \cdot \log_{10} 3 \]

### Step 2: Substitute the value of \(\log_{10} 3\)
From the problem, we know that \(\log_{10} 3 = 0.477\). Substituting this value in, we get:

\[ \log_{10} (3^{40}) = 40 \cdot 0.477 \]

### Step 3: Calculate \(40 \cdot 0.477\)
Now, we perform the multiplication:

\[ 40 \cdot 0.477 = 19.08 \]

### Step 4: Find the floor value
Next, we take the floor of \(19.08\):

\[ \lfloor 19.08 \rfloor = 19 \]

### Step 5: Calculate the number of digits
Finally, we substitute this into the formula for the number of digits:

\[ d = \lfloor \log_{10} (3^{40}) \rfloor + 1 = 19 + 1 = 20 \]

Thus, the number of digits in \(3^{40}\) is \(20\).

### Final Answer
The number of digits in \(3^{40}\) is \(20\).

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