# Calculations using googolplexes

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?

• What makes you think that first inequality holds? What do you know about logarithms? Commented Dec 19, 2014 at 23:07
• Depending on how exact of an answer you want, $\frac{10^{(10^{100})}}{10^{(10^{70})}}\approx 10^{10^{100}}$ :) Commented Dec 19, 2014 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. Commented Dec 19, 2014 at 23:14
• turkeyhundt - I guess I am struggling to comprehend the magnitude of the numbers :) Commented Dec 19, 2014 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. Commented Dec 19, 2014 at 23:18

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.)

• 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. Commented Dec 19, 2014 at 23:49

$$\frac{10^{10^{100}}}{10^{10^{70}}}=10^{10^{100}-10^{70}}=10^{10^{70}(10^{30}-1)}$$ 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."

• Thanks Matt, Can you explain how you get $10^{30} - 1$ please? Commented Dec 19, 2014 at 23:20
• @Jayprakash I factored out $10^{70}$ from $10^{100}-10^{70}$. Commented Dec 19, 2014 at 23:21
• You can factor out a $10^{70}$ from the exponent Commented Dec 19, 2014 at 23:22
• Cheers, thanks guys Commented Dec 19, 2014 at 23:24
• You're welcome. Commented Dec 19, 2014 at 23:25

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