The problem is NOT with LOG function.

You are using INTEGER DIVISION!

@underlying and @strike are both integers.

select @d1 = (Log(@underlying / @strike) + @RiskFree * @expTime) / (@vol * SQRT(@expTime)) + 0.5 * @vol * SQRT(@expTime)

Try this

select @d1 = (Log(1.0 * @underlying / @strike) + @RiskFree * @expTime) / (@vol * SQRT(@expTime)) + 0.5 * @vol * SQRT(@expTime)