Question: How to convert a number to $10^n$?

For example, $9.3\times10^{-6}=10^{-5.03}$.

How do you do this conversion to calculate the exponent of $10$?

How does one calculate the $-5.03$ exponent?

Answer: 
Well... logarithm values are not easy to hand-calculate, if that's what you're asking. Because $x = 10^{\log\_{10} x}$ is the simple definition of logarithm values.

Which is why, once upon a time, kids like me were taught how to look up the relevant values (for $9.3$, in your case) in a table of logarithms: $0.9685$, and then calculate the final logarithm by adjusting for the power of ten. For ease of calculation in later steps, the integer value was sometimes shown negative with the fractional part positive - so your answer there would be $\bar 6.9685$, meaning $-6+0.9685 = -5.0315$ .
Just put it in logarithmic form, since expression is $a^n$.

$\log \_{10}(x) = n$

$x = 10^n$

where

$n = \log\_{10}(9,3 \times 10^{-6}) = x$