To answer these questions without the aid of a calculator, you have understand some basic definitions and rules. To get a LOG (and this is the LOG in base 10, because when the base is unstated, we set it to 10), the question you have answer is: what power do I need to raise 10 to, to get the number you want the log of?
For instance in a) you want LOG (0.1) Since 10^(-1) is 0.1, the LOG of 0.1 is (-1). In c) you want the LOG of 100. 100 = 10^2, so the LOG of 100 is 2. Similarly the LOG of 1000 is 3, because 10 to the power 3 is 1000.
The above questions you pesent could be answered without the aid of a calculator with just that information. However, understanding how the rules of exponents and logs are related, would further simply these equations, making them even easier to solve without the aid of a calculator. For instance:
5 LOG 100 = (substitute 100 = 10 squared) =
5 LOG 10^2 = (use the "power rule" of logs, in reverse) =
LOG [(10^2)^5] = (when taking a power to a power, multiply the exponents) =
LOG (10^10) = (definition of log) =
10
Thoroughly reviewing the rules of logs and exponents will help greatly, and provide even more simplifications.
Qestion d) requires more thought. For d), once you understand that the LOG is the power you need to raise the number 10 to, to get the number you are looking for the log of - - - you will see that no matter what the power is on 10, since 10 is positive, if I keep multiply it, I will never get a negative number! That is why the LOG of a negative number is undefined.
Good questions!
Good luck!
-Diane
Parviz F.
02/13/14