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?
On
$$\frac{10^{10^{100}}}{10^{10^{70}}} = 10^{(10^{100}-10^{70})} \\ = 1, \text{ followed by 10 duovigintillion less than 1 googol zeroes}$$
On
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.)
$$\frac{10^{10^{100}}}{10^{10^{70}}}=10^{10^{100}-10^{70}}=10^{10^{70}(10^{30}-1)}$$ 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."