We can evaluate 999log(9) = 953.288
This equals about
(10953)*10.288 =
10953 * 1.942 =
1.942 x 10953
All in all......a really big number.....1942 followed by 950 zeros.......!!! {more or less}
Thanks Chris,
I am only just getting the hang of these.
It is a tricky way of finding high powers that the calc cannot find on its own.
If you can do a calc that the calc cannot do on its own does that kind of make you = 42?
What is 9 to the power of 999 ?
change of basis: 9 to 10
$$b_1^{ e_1 } = b_2^{ e_2 } \quad | \quad \ln() \\
e_1 \cdot \ln{(b_1)} = e_2 \cdot \ln{(b_2)} \\\\
e_2 = e_1 \cdot \dfrac{ \ln{(b_1)} } { \ln{(b_2)} } \\
\boxed{
b_1^{ e_1 } = b_2^{ e_1
\left(
\cdot \dfrac{ \ln{(b_1)} } { \ln{(b_2)} }
\right)
}}\\\\
\small{\text{
$
b_1 = 9 \qquad e_ 1 = 999 \qquad b_2 = 10
$
}} \\\\
\small{\text{
$
9^{999} = 10^{999
\left(
\cdot \dfrac{ \ln{(9)} } { \ln{(10)} }
\right)
}
$
}} \\\\
\small{\text{
$ 9^{999} = 10^{999 \cdot( 0.95424250944 ) } $ }} \\
\small{\text{
$ 9^{999} = 10^{953.288266930} $
}} \\
\small{\text{
$ 9^{999} = 10^{0.288266930}\cdot 10^{953} $
}} \\
\small{\text{
$ 9^{999} = 1.94207916858
\cdot 10^{953} $
}}$$