Unit 6

Exponential and Logarithmic Functions

Lesson 2: Review of Logarithms and Exponential Functions

Familiarizing ourselves with the simplification and the expansion of logarithmic expressions will help us become more comfortable using the laws of logarithms. And, in some cases, it is more helpful to expand a logarithmic expression than it is to simplify.

Example 1

Expand «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«msub»«mi»log«/mi»«mn»7«/mn»«/msub»«mfenced»«mfrac»«mrow»«msup»«mi»x«/mi»«mn»5«/mn»«/msup»«mi»y«/mi»«/mrow»«msqrt»«mi»z«/mi»«/msqrt»«/mfrac»«/mfenced»«/mrow»«/mstyle»«/math».

Solution

Three of the logarithmic laws are used, the quotient law, the product law, and the power law.

First, expand by applying the quotient law of logarithms.

«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mo»=«/mo»«msub»«mi»log«/mi»«mn»7«/mn»«/msub»«mfenced»«mrow»«msup»«mi»x«/mi»«mn»5«/mn»«/msup»«mi»y«/mi»«/mrow»«/mfenced»«mo»-«/mo»«msub»«mi»log«/mi»«mn»7«/mn»«/msub»«mfenced»«msqrt»«mi»z«/mi»«/msqrt»«/mfenced»«/mrow»«/mstyle»«/math»

Recall that «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«msqrt»«mi»z«/mi»«/msqrt»«mo»=«/mo»«msup»«mi»z«/mi»«mfrac»«mn»1«/mn»«mn»2«/mn»«/mfrac»«/msup»«/mrow»«/mstyle»«/math».

The expression is now written as follows.

«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mo»=«/mo»«msub»«mi»log«/mi»«mn»7«/mn»«/msub»«mfenced»«mrow»«msup»«mi»x«/mi»«mn»5«/mn»«/msup»«mi»y«/mi»«/mrow»«/mfenced»«mo»-«/mo»«msub»«mi»log«/mi»«mn»7«/mn»«/msub»«mfenced»«msup»«mi»z«/mi»«mfrac»«mn»1«/mn»«mn»2«/mn»«/mfrac»«/msup»«/mfenced»«/mrow»«/mstyle»«/math»

Now, expand by applying the product law.

«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mo»=«/mo»«msub»«mi»log«/mi»«mn»7«/mn»«/msub»«mfenced»«msup»«mi»x«/mi»«mn»5«/mn»«/msup»«/mfenced»«mo»+«/mo»«msub»«mi»log«/mi»«mn»7«/mn»«/msub»«mfenced»«mi»y«/mi»«/mfenced»«mo»-«/mo»«msub»«mi»log«/mi»«mn»7«/mn»«/msub»«mfenced»«msup»«mi»z«/mi»«mfrac»«mn»1«/mn»«mn»2«/mn»«/mfrac»«/msup»«/mfenced»«/mstyle»«/math»

And, simplify further using the power law.

«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mo»=«/mo»«mn»5«/mn»«msub»«mi»log«/mi»«mn»7«/mn»«/msub»«mfenced»«mi»x«/mi»«/mfenced»«mo»+«/mo»«msub»«mi»log«/mi»«mn»7«/mn»«/msub»«mfenced»«mi»y«/mi»«/mfenced»«mo»-«/mo»«mfrac»«mn»1«/mn»«mn»2«/mn»«/mfrac»«msub»«mi»log«/mi»«mn»7«/mn»«/msub»«mfenced»«mi»z«/mi»«/mfenced»«/mrow»«/mstyle»«/math»

Evaluating Expressions Using Logarithmic Laws

Use your understanding of the relationship between logarithms and exponents, along with the laws of logarithms, logarithmic expressions can be evaluated.

Example 2

Use logarithmic laws to evaluate each expression.

«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«msub»«mi»log«/mi»«mn»3«/mn»«/msub»«mn»9«/mn»«msqrt»«mn»3«/mn»«/msqrt»«/mrow»«/mstyle»«/math»

Solutions

«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mtable columnalign=¨center left¨»«mtr»«mtd»«mo»=«/mo»«/mtd»«mtd»«msub»«mi»log«/mi»«mn»3«/mn»«/msub»«mn»9«/mn»«mo»+«/mo»«msub»«mi»log«/mi»«mn»3«/mn»«/msub»«msqrt»«mn»3«/mn»«/msqrt»«/mtd»«/mtr»«mtr»«mtd»«mo»=«/mo»«/mtd»«mtd»«msub»«mi»log«/mi»«mn»3«/mn»«/msub»«msup»«mn»3«/mn»«mn»2«/mn»«/msup»«mo»+«/mo»«msub»«mi»log«/mi»«mn»3«/mn»«/msub»«msup»«mn»3«/mn»«mfrac»«mn»1«/mn»«mn»2«/mn»«/mfrac»«/msup»«/mtd»«/mtr»«mtr»«mtd»«mo»=«/mo»«/mtd»«mtd»«mn»2«/mn»«msub»«mi»log«/mi»«mn»3«/mn»«/msub»«mn»3«/mn»«mo»+«/mo»«mfrac»«mn»1«/mn»«mn»2«/mn»«/mfrac»«msub»«mi»log«/mi»«mn»3«/mn»«/msub»«mn»3«/mn»«/mtd»«/mtr»«/mtable»«/mstyle»«/math»

Now, recall that «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«msub»«mi»log«/mi»«mi»c«/mi»«/msub»«mo»=«/mo»«mn»1«/mn»«/mrow»«/mstyle»«/math», so «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«msub»«mi»log«/mi»«mn»3«/mn»«/msub»«mo»=«/mo»«mn»1«/mn»«/mrow»«/mstyle»«/math».

«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mn»2«/mn»«msub»«mi»log«/mi»«mn»3«/mn»«/msub»«mn»3«/mn»«mo»+«/mo»«mfrac»«mn»1«/mn»«mn»2«/mn»«/mfrac»«msub»«mi»log«/mi»«mn»3«/mn»«/msub»«mn»3«/mn»«mspace linebreak=¨newline¨/»«mo»=«/mo»«mn»2«/mn»«mfenced»«mn»1«/mn»«/mfenced»«mo»+«/mo»«mfrac»«mn»1«/mn»«mn»2«/mn»«/mfrac»«mfenced»«mn»1«/mn»«/mfenced»«mspace linebreak=¨newline¨/»«mo»=«/mo»«mn»2«/mn»«mfrac»«mn»1«/mn»«mn»2«/mn»«/mfrac»«/mrow»«/mstyle»«/math»

«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»2«/mn»«msub»«mi»log«/mi»«mn»3«/mn»«/msub»«mn»6«/mn»«mo»-«/mo»«mfrac»«mn»1«/mn»«mn»2«/mn»«/mfrac»«msub»«mi»log«/mi»«mn»3«/mn»«/msub»«mn»64«/mn»«mo»+«/mo»«msub»«mi»log«/mi»«mn»3«/mn»«/msub»«mn»2«/mn»«mspace linebreak=¨newline¨/»«mo»=«/mo»«msub»«mi»log«/mi»«mn»3«/mn»«/msub»«msup»«mn»6«/mn»«mn»2«/mn»«/msup»«mo»-«/mo»«msub»«mi»log«/mi»«mn»3«/mn»«/msub»«msup»«mn»64«/mn»«mfrac»«mn»1«/mn»«mn»2«/mn»«/mfrac»«/msup»«mo»+«/mo»«msub»«mi»log«/mi»«mn»3«/mn»«/msub»«mn»2«/mn»«mspace linebreak=¨newline¨/»«mo»=«/mo»«msub»«mi»log«/mi»«mn»3«/mn»«/msub»«mn»36«/mn»«mo»-«/mo»«msub»«mi»log«/mi»«mn»3«/mn»«/msub»«msqrt»«mn»64«/mn»«/msqrt»«mo»+«/mo»«msub»«mi»log«/mi»«mn»3«/mn»«/msub»«mn»2«/mn»«mspace linebreak=¨newline¨/»«mo»=«/mo»«msub»«mi»log«/mi»«mn»3«/mn»«/msub»«mfenced»«mfrac»«mn»36«/mn»«mn»8«/mn»«/mfrac»«/mfenced»«mfenced»«mn»2«/mn»«/mfenced»«mspace linebreak=¨newline¨/»«mo»=«/mo»«msub»«mi»log«/mi»«mn»3«/mn»«/msub»«mfenced»«mfrac»«mn»72«/mn»«mn»8«/mn»«/mfrac»«/mfenced»«mspace linebreak=¨newline¨/»«mo»=«/mo»«msub»«mi»log«/mi»«mn»3«/mn»«/msub»«mfenced»«mn»9«/mn»«/mfenced»«/mstyle»«/math»

Now, let «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»x«/mi»«mo»=«/mo»«msub»«mi»log«/mi»«mn»3«/mn»«/msub»«mfenced»«mn»9«/mn»«/mfenced»«/mrow»«/mstyle»«/math».

«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mtable columnalign=¨right center left¨»«mtr»«mtd»«msup»«mn»3«/mn»«mi»x«/mi»«/msup»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mn»9«/mn»«/mtd»«/mtr»«mtr»«mtd»«msup»«mn»3«/mn»«mi»x«/mi»«/msup»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«msup»«mn»3«/mn»«mn»2«/mn»«/msup»«/mtd»«/mtr»«mtr»«mtd»«mi»x«/mi»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mn»2«/mn»«/mtd»«/mtr»«/mtable»«/mstyle»«/math»

Reminder: The laws of logarithms and the laws of exponents can only be used when the logarithms or the exponents have the same base.

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U6 L2 Laws of Logarithms Skill Builder