8.4 – Properties of Logarithms

Properties of Logarithms There are four basic properties of logarithms that we will be working with. For every case, the base of the logarithm can not be equal to 1 and the values must all be positive (no negatives in logs)

inverses “undo” each each other Since logs and exponentials of the same base are inverse functions of each other they “undo” each other. Remember that: This means that: inverses “undo” each each other = 7 = 5

Product Rule logbMN = LogbM + logbN Ex: logbxy = logbx + logby log 2 + log 3 log39 + log3b

Quotient Rule Ex:

Power Rule Ex:

= = = = Properties of Logarithms 1. 2. 3. CONDENSED EXPANDED (these properties are based on rules of exponents since logs = exponents)

Let’s try some Working backwards now: write the following as a single logarithm.

Let’s try some Write the following as a single logarithm.

Let’s try something more complicated . . . Condense the logs log 5 + log x – log 3 + 4log 5

Using the log properties, write the expression as a sum and/or difference of logs (expand). When working with logs, re-write any radicals as rational exponents. using the second property: using the first property: using the third property:

Using the log properties, write the expression as a single logarithm (condense). using the third property: this direction using the second property: this direction

More Properties of Logarithms This one says if you have an equation, you can take the log of both sides and the equality still holds. This one says if you have an equation and each side has a log of the same base, you know the "stuff" you are taking the logs of are equal.

Let’s try something more complicated . . . Condense the logs log 5 + log x – log 3 + 4log 5

Let’s try something more complicated . . . Expand

Let’s try something more complicated . . . Expand