Understanding the Number of Digits in X³

The problem asks us to determine the number of digits in the number $X³$, given that $X$ is a 30-digit number starting with the digits 47.

Estimating the Value of X

A 30-digit number is any integer $N$ such that $10^{29} \le N < 10^{30}$. Since $X$ is a 30-digit number starting with 47, we know its value is roughly $4.7 \times 10^{29}$. More precisely, $X$ lies in the range:

$$4.7 \times 10^{29} \le X < 4.8 \times 10^{29}$$

Calculating the Number of Digits Using Logarithms

The number of digits in a positive integer $N$ can be calculated using the base-10 logarithm: Number of digits = $\lfloor \log_{10}(N) \rfloor + 1$.

In our case, we need to find the number of digits in $X³$. So, we need to calculate $\lfloor \log_{10}(X³) \rfloor + 1$.

Using the properties of logarithms, $\log_{10}(X³) = 3 \times \log_{10}(X)$.

Therefore, the number of digits in $X³$ is $\lfloor 3 \times \log_{10}(X) \rfloor + 1$.

Finding the Range for $\log_{10}(X³)$

Let's find the range for $3 \times \log_{10}(X)$ using the estimated range for $X$:

Lower Bound:

If $X = 4.7 \times 10^{29}$, then

$$ \log_{10}(X) = \log_{10}(4.7 \times 10^{29}) = \log_{10}(4.7) + \log_{10}(10^{29}) $$

$$ \log_{10}(X) = \log_{10}(4.7) + 29 $$

Since $\log_{10}(4.7)$ is approximately $0.672$,

$$ \log_{10}(X) \approx 0.672 + 29 = 29.672 $$

Now, multiply by 3:

$$ 3 \times \log_{10}(X) \approx 3 \times 29.672 = 89.016 $$

Upper Bound:

If $X$ is just below $4.8 \times 10^{29}$, then

$$ \log_{10}(X) < \log_{10}(4.8 \times 10^{29}) = \log_{10}(4.8) + 29 $$

Since $\log_{10}(4.8)$ is approximately $0.681$,

$$ \log_{10}(X) < 0.681 + 29 = 29.681 $$

Now, multiply by 3:

$$ 3 \times \log_{10}(X) < 3 \times 29.681 = 89.043 $$

So, we have the range:

$$ 89.016 \le 3 \times \log_{10}(X) < 89.043 $$

Determining the Number of Digits

The number of digits in $X³$ is $\lfloor 3 \times \log_{10}(X) \rfloor + 1$.

Using the calculated range, the floor value is:

$$ \lfloor 3 \times \log_{10}(X) \rfloor = \lfloor \text{a number between } 89.016 \text{ and } 89.043 \rfloor = 89 $$

Therefore, the number of digits is:

$$ 89 + 1 = 90 $$

Conclusion

Based on the logarithmic calculation, the number $X³$ will have 90 digits.