There are a few options, the simplest is to conduct 2 SUMIFS... let's assume as before H24 has the "From" date and I24 has the "To" date then to get the Sum of values in between those dates:
=SUMIF('Expenses Log'!$C$8:$C$100,">="&H24,'Expenses Log'!$I$8:$I$200)-SUMIF('Expenses Log'!$C$8:$C$100,">"&I24,'Expenses Log'!$I$8:$I$200)
The above sums value >= H24 and subtracts values that exceed I24.
The other common approach is to use a Sumproduct though this is arguably more "expensive" than the SUMIF approach.
EDIT: apologies JB, simultaneous posts strike again...
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