# How many digits are there in $2^{17}\times 3^2\times 5^{14}\times 7 ?$

How many digits are there in $$2^{17}\times 3^2\times 5^{14}\times 7 ?$$

I agree with the fellow who asked that if one cannot have 2 and 5 in the number above how we will calculate the number of digits???

• According to WolframAlpha, there are $17$ digits. wolframalpha.com/input/?i=2%5E17%C3%973%5E2%C3%975%5E14%C3%977 Commented Jun 24, 2014 at 4:33
• $\left\lceil 17\log_{10}2 + 2\log_{10}3 + 14\log_{10}5 + \log_{10}7\right\rceil$. Commented Jun 24, 2014 at 4:34
• without knowing log, we could not calculate in this method.. right?@Oleg567 Commented Jul 2, 2014 at 5:59

See if I multiply $2$ and $5$, I will get $10$. So $2^{14}$ and $5^{14}$ when multpilied will give $10^{14}$ which has 14 zeroes. All that remains to be multiplied is $8$ , $9$ and $7$, which is three digits when done. I already had $14$ digits. In total $17$ digits

• I'm not the asker, so I dont know if OP is interested, but is there some interesting theory regarding this? How about if one doensnt have 5 and 2 as a factor? Then you can't just count the number of tens. How does it work then? Commented Jun 24, 2014 at 10:17
• @JuliusL33t .. u can find something related to your query here www.math.stackexchange.com/questions/644889/calculate-the-number-of-digits-in-a-product-of-large-numbers Commented Jun 24, 2014 at 10:25
• oh super... thankyu somuch @TattwamasiAmrutam Commented Jul 2, 2014 at 5:56

If we selectively combine terms as we evaluate:

$$2^{17} \times 3^2 \times 5^{14} \times 7 = 10^{14} \times 2^3 \times 3^2 \times 7$$

$$= 10^{14} \times 504$$

In particular, $$10^{16} < (10^{14} \times 504) < 10^{17}$$

I'll let you fill in the details. :)