Evaluate the following: Recall: a logarithm is an exponent. So in each case, we are looking for the exponent of 2 to get a number. In the first example, the answer is 1 Because

Compare example 4 with examples 2 and 3 What do you notice? If we multiply 4 and 16 we’ll get 64 If we add 2 and 4 we get 6 There is a relationship between these examples.

Another way of seeing this: But we know that So we can substitute to get: So we found that: We should check our answer: This is true, so our answer is correct.

In the same manner, we can create rules for the logarithm of a quotient and the logarithm of something raised to a power If we do that, we have the following properties of logarithms 1) The product rule: 2) The quotient rule: 3) The power rule:

Example – Use the laws of logarithms to expand the logarithm There are two laws we can apply here: the power and the product rule. We must do the product rule first. Now we can apply the power rule:

Example – Use the laws of logarithms to expand the logarithm Quotient rule first We can apply the power rule also, if we change the form of the square root of x

Example – Use the laws of logarithms to write the following expression as a single logarithm Before we use quotient rule, we must use the power rule Now the quotient rule

Example – Use the laws of logarithms to write the following expression as a single logarithm