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## 6.5 Properties of Logarithms

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**6.5 Properties of Logarithms**In this section, we will study the following topics: Using the properties of logarithms to evaluate log expressions Using the properties of logarithms to expand or condense log expressions Using the change-of-base formula**Properties of Logarithmic Functions**To solve various types of problems involving logarithms, we can make use of the following properties of logarithms.**Properties of Logarithms**The four properties of logarithms listed earlier hold true for natural logarithms as well.**Examples**• Find the exact value of each expression without using a calculator.**Properties of Logarithms**WARNING!!!!!!**Properties of Logarithms**ANOTHER WARNING!!!!!!**Using the Properties of Logarithms to Expand Log Expressions**Use the properties of logs to write the expression as the sum and/or difference of logarithms (EXPAND).**Using the Properties of Logarithms to Expand Log Expressions****Using the Properties of Logarithms to Expand Log Expressions****Using the Properties of Logarithms to Expand Log Expressions****Using the Properties of Logarithms to Condense Log**Expressions Use the properties of logs to write each expressions as a single logarithm (CONDENSE).**Using the Properties of Logarithms to Condense Log**Expressions**Using the Properties of Logarithms to Condense Log**Expressions**Change-of-Base Formula**• I mentioned in a previous section that the calculator only has two types of log keys: • COMMON LOG (BASE 10) • NATURAL LOG (BASE e). • It’s true that these two types of logarithms are used most often, but sometimes we want to evaluate logarithms with bases other than 10 or e.**Change-of-Base Formula**To do this on the calculator, we use a CHANGE OF BASE FORMULA. We will convert the logarithm with base a into an equivalent expression involving common logarithms or natural logarithms.**Change-of-Base Formula Examples**Example*: Use the change-of-base formula to evaluate log7264 using common logarithms using natural logarithms. Solution: The result is the same whether you use the common log or the natural log.**Change-of-Base Formula Examples**Example: Use the change-of-base formula to evaluate: a) b)