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How can I calculate $\dfrac{10^{10^{100 }}}{ 10^{10^{70}}}$?

I have tried using logs ie:

$$\frac{10^{10^{100}}}{10^{10^{70}}}$$

$$=\frac{(100\times \ln(10)) \times \ln(10)}{(70\times \ln(10)) \times \ln(10)}$$

$$=\frac{10}{7}$$

which looks incorrect as $\dfrac{10^{10}}{10^7}=1000$

What am I doing wrong?

Edward Jiang
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Jayprakash
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  • What makes you think that first inequality holds? What do you know about logarithms? – Hayden Dec 19 '14 at 23:07
  • Depending on how exact of an answer you want, $\frac{10^{(10^{100})}}{10^{(10^{70})}}\approx 10^{10^{100}}$ :) – turkeyhundt Dec 19 '14 at 23:09
  • Hayden - not sure what you mean by inequality but I am assuming that if x^y = y ln x then x^y^z = (z ln y) ln x. – Jayprakash Dec 19 '14 at 23:14
  • turkeyhundt - I guess I am struggling to comprehend the magnitude of the numbers :) – Jayprakash Dec 19 '14 at 23:15
  • I know. It is difficult. I had to start over in my thinking a few times. Matt gives a good answer below. – turkeyhundt Dec 19 '14 at 23:18
  • To get some comprehension of how much bigger $10^{10^{100}}$ is than $10^{10^{70}}$ have a look at this. You are essentially working out $10^{100} - 10^{70} = 10^{70}(10^{30} -1)$ Then take log of this $70\ln(10) + \ln{(10^{30} -1)}$, its pretty easy to see $(10^{30} -1)$ is very very close to just $10^{30}$ so if we replace it with that our equation goes to $70\ln(10) + 30\ln(10) = 100\ln(10)$. So $10^{100} - 10^{70} \approx 10^{100}$. – Rammus Dec 19 '14 at 23:30

3 Answers3

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As the other answers have said, there's no real simplifications you can do, but I wanted to add that $\log(a^{b^c})$ is not $c \cdot \log(a) \cdot \log(b)$. Instead, $\log(a^{b^c}) = b^c \cdot \log(a)$. (That is, the log of a googolplex is about 2.3 googol.)

If you take the log of that, you get $\log(\log(a^{b^c})) = c \cdot \log(b \cdot \log(a))$, but taking the log of the log of a ratio isn't as useful as just taking the log. ($\log(\log(a/b)) = \log(\log(a) - \log(b))$, which you can't do much more with in the general case.)

Joshua Taylor
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    I wasn't sure how to apply the rule to exponents of exponents and the internet wasn't too forthcoming using generic searches. Thankfully there are site like this one. – Jayprakash Dec 19 '14 at 23:49
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$$\frac{10^{10^{100}}}{10^{10^{70}}}=10^{10^{100}-10^{70}}=10^{10^{70}(10^{30}-1)}$$ There's more you can do but it doesn't really get any simpler than that. It's $1$ with $10^{70}(10^{30}-1)$ zeros after it. You can turn the $10^{30}-1$ into a string of 30 nines, but I don't see any particular purpose to that. You should read the exponent as "way too many zeros to write down."

Matt Samuel
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$$\frac{10^{10^{100}}}{10^{10^{70}}} = 10^{(10^{100}-10^{70})} \\ = 1, \text{ followed by 10 duovigintillion less than 1 googol zeroes}$$

John
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