# What are the rules for natural logarithms

### Calculation rules for the logarithm

The logarithm calculation rules allow the simplification of arithmetic operations and are therefore often the reason for the introduction and treatment of the logarithm. The following overview shows how the arithmetic operations are "reduced" by the transition to arithmetic with logarithms: The logarithm concept is based on the power concept, which is linked to a wealth of rules (see accompanying text "Powers and exponential functions). So it's no wonder that we have to refer to these rules to understand the logarithm calculation rules. For the sake of brevity, we only want them for the Prove the (particularly important) decadic logarithm. In addition, we note the corresponding laws for the natural and the general logarithm.
Consequences from the logarithm definition
Rule 2: The following apply: Also: Logarithms of products and quotients
What can one say about the logarithm of the product of two numbers? We discover the rule using a concrete example. Let's first look at Figure 4668 with the function , which is shown enlarged in the number range between 0 and 1 for better clarity. Fig. 4668 The function y = 10 ^ (x) in the range x = 0 to x = 1
Rule 3:  Exercise: A similar rule applies to the logarithm of a quotient. Rule 3 shows that the multiplication becomes an addition by moving to the logarithm. Analogously, one finds that when calculating with the logarithm of a quotient, the division is transformed into a subtraction. The structure of the proof is completely identical to that in the chapter "Proofs". If you want, you can try your hand at the proof by modifying steps 1 through 5 to prove rule 3 as appropriate. One then obtains:
Rule 4:  Conclusion:  Exercise: The logarithm of a power
What can be said about the logarithm of a power with any real exponent? If we use the findings from rules 3 and 4 ("times" becomes "plus", "divided" becomes "minus"), one could expect that raising the power would result in a multiplication, so that: And in fact, the evidence for this presumption in the "Demonstration of evidence" chapter confirms the following rule:
Rule 5:  Exercise: If the base of the power is the same as the base of the logarithm, rule 5 simplifies further because: According to rule 2, however, applies , so: Conclusion:  Conclusion:  Finally - in order to avoid confusion - it should be mentioned that the expression can not be further simplified. When calculating with logarithms, quite complicated expressions can arise, but some of them can be simplified considerably. The following relationship will be of great help to you: This equation is actually nothing more than applications of definition 2 and rule 1:
So in general we get:
Rule 6:  Exercise: 