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Properties of Logarithms. Section 3.3. Properties of Logarithms. What logs can we find using our calculators? Common logarithm Natural logarithm Although these are the two most frequently used logarithms, you may need to evaluate other logs at times - PowerPoint PPT Presentation

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Properties of Logarithms

Properties of LogarithmsSection 3.3Properties of LogarithmsWhat logs can we find using our calculators?Common logarithmNatural logarithm

Although these are the two most frequently used logarithms, you may need to evaluate other logs at times

For these instances, we have a change-of-base formulaProperties of LogarithmsChange-of-Base Formula

Let a, b, and x be positive real numbers such that a 1 and b 1. Then can be converted to a different base as follows:

Base bProperties of LogarithmsChange-of-Base Formula

Let a, b, and x be positive real numbers such that a 1 and b 1. Then can be converted to a different base as follows:

Base 10Properties of LogarithmsChange-of-Base Formula

Let a, b, and x be positive real numbers such that a 1 and b 1. Then can be converted to a different base as follows:

Base eProperties of LogarithmsEvaluate the following logarithm:

4 raised to what power equals 30?Since we dont know the answer to this, we would want to use the change-of-base formula

Properties of LogarithmsEvaluate the following logarithm using the natural log function.

Properties of LogarithmsEvaluate the following logarithms using the common log and the natural log.

Properties of LogarithmsWhat is a logarithm?

Therefore, logarithms should have properties that are similar to those of exponentsProperties of LogarithmsFor example, evaluate the following:

Properties of LogarithmsJust like we have properties for exponents, we have properties for logarithms.

These properties are true for logs with base a, the common logs, and the natural logsProperties of LogarithmsProperties of LogarithmsLet a be a positive number such that a 1, and let n be a real number. If u and v are positive real numbers, the following properties are true.

Properties of LogarithmsUse the properties to rewrite the following logarithm:

From property 1, we can rewrite this as the following:

Properties of LogarithmsUse the properties to rewrite the following logarithm:

From property 2, we can rewrite this as the following:

Properties of LogarithmsUse the properties to rewrite the following logarithm:

From property 3, we can rewrite this as the following:

Properties of LogarithmsSection 3.3Properties of LogarithmsYesterday:

Change-of-Base Formula

3 PropertiesProperties of LogarithmsToday we are going to continue working with the three properties covered yesterday.

Properties of LogarithmsThese properties can be used to rewrite log expressions in simpler terms

We can take complicated products, quotients, and exponentials and convert them to sums, differences, and productsProperties of LogarithmsExpand the following log expression:

Start by applying property 1 to separate the product:

Properties of LogarithmsExpand the following log expression:Apply property 3 to eliminate the exponent

Properties of LogarithmsExpand the following expression:

Start by applying property 1 to separate the product:

Properties of LogarithmsExpand the following expression:Eliminate the exponents

Properties of LogarithmsRewrite the following logarithm:

For problems involving square roots, begin by converting the square root to a power

Properties of Logarithms

Apply property 1 to get rid of the quotient:

Properties of Logarithms

Apply property 3 to get rid of the exponent

Properties of LogarithmsRewrite the following logarithmic expressions:

Properties of LogarithmsExpand the following expression:

Properties of Logarithms

Properties of LogarithmsSection 3.3Properties of LogarithmsSo far in this section, we have:

Change-of-Base Formula3 PropertiesExpanded expressionsToday we are going to do the exact oppositeCondense expressions

Properties of LogarithmsWhen we were expanding, what order did we typically apply the properties in? Property 1 or Property 2End with Property 3

When we condense, we use the opposite orderProperty 3Property 1 or Property 2

Properties of LogarithmsThe most common error:

Log x Log y

When you condense, you are condensing the expression down to one log function

Properties of LogarithmsCondense the following expression:

Start by applying property 3, then move on to properties 1 and 2

Properties of Logarithms

Is this expression simplified to one log function?Properties of LogarithmsCondense the following expression:

Properties of Logarithms

Properties of Logarithms

Properties of Logarithms

Properties of Logarithms

Properties of LogarithmsProperties of LogarithmsProperties of LogarithmsProperties of LogarithmsProperties of Logarithms