Number of digits in N = 210 × 72 × 35

To find the number of digits in N, we need to determine the value of N first by multiplying the given numbers.

Step 1: Calculate the value of N
N = 210 × 72 × 35
N = 529200 × 35
N = 18,517,200

Step 2: Find the number of digits in N
To determine the number of digits in N, we can take the logarithm of N to the base 10 and add 1 to the result. This will give us the total number of digits in N.

Using the logarithmic properties, we can simplify the calculation as follows:

log(N) = log(18,517,200)
log(N) = log(529200) + log(35)
log(N) = log(2^4 × 5^2 × 5292) + log(5 × 7)
log(N) = 4log(2) + 2log(5) + log(5292) + log(5) + log(7)
log(N) = 4(0.301) + 2(0.477) + log(5292) + log(5) + log(7)
log(N) = 1.204 + 0.954 + log(5292) + log(5) + log(7)
log(N) = 2.158 + log(5292) + log(5) + log(7)

Now, we need to determine the values of log(5292), log(5), and log(7) using the given information.

Given:
log(5) = 0.301
log(7) = 0.845

To find log(5292), we can break it down further:
5292 = 2^2 × 3 × 7^2 × 7

Using logarithmic properties, we can simplify the calculation as follows:
log(5292) = log(2^2 × 3 × 7^2 × 7)
log(5292) = 2log(2) + log(3) + 2log(7) + log(7)
log(5292) = 2(0.301) + log(3) + 2(0.845) + log(7)
log(5292) = 0.602 + log(3) + 1.69 + log(7)
log(5292) = 2.292 + log(3) + log(7)

Now, substituting the values back into the previous equation:
log(N) = 2.158 + 2.292 + log(3) + log(7) + 0.301 + 0.845
log(N) = 5.596 + log(3) + log(7)

Step 3: Calculate the value of N
To determine the value of N, we need to convert the logarithmic expression back into its exponential form.

N = 10^(log(N))
N = 10^(5.596 + log(3) + log(7))

Now, we can calculate N using the given values of log(3) and log(7