First, subtracting 5 from all sides:
4 < 10^x + 5 < 9
4-5 < 10^x + 5 - 5 < 9-5
-1 < 10^x < 4
We can take the log_10 of all the sides to bring down the x:
log_10(-1) < x < log_10(4)
However, log_10(-1) is undefined; log_10(x) only exists for x
ge 0. So, the original question is kind of misleading; 10^x+5 only gives values for x ge 5, so it would be unnecessary to put 4 on the left hand side.
Since log_10(x) goes towards -\infty as x\rightarrow0, we can just replace log_10(-1) with -infty:
-infty < x < log_10(4)
The -infty is now redundant, because that doesn't place any lower bound on x, so we can remove that entirely, giving us our answer:
x < log_10(4)
x < 0.60206