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Identifying and Graphing Log Functions
Section 3.2 Precalculus PreAP/Dual, Revised Β©2017 11/24/2018 8:03 PM 3.2: Log Functions
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Real-Life Situation The pH scale is used in chemistry to determine the acidity or alkalinity of a solution. The scale ranges from π to ππ, with π being the most acidic and ππ the most alkaline The difference in strength of an acid of ππ― π and that of ππ― π is not twofold, but tenfold The pH scale is actually a logarithm in the form: πππππ, thus, ππ― π = πππππππ, and ππ― π = ππππππππ 11/24/2018 8:03 PM 3.2: Log Functions
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Real Life Situation The Richter scale is used to determine the strength of the ground movement. The larger the number, the more violent the movement Similar to the pH scale, an earthquake of magnitude 7 on the scale is ten times stronger than an earthquake of magnitude 6. Again, this is because the Richter scale is actually a logarithm: log10[measurement of movement of the earth] 11/24/2018 8:03 PM 3.2: Log Functions
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Real-Life Situation The Modified Richter Scale uses a modified scale.
It is not ten-fold A separate equation is used Music βSemitonesβ The interval between two notes in semitones is the base-21/12Β logarithm of the frequency ratio (or equivalently, 12 times the base-2 logarithm). Astronomy The magnitude measures the starsβ brightness logarithmically with vision Source: Wikipedia 11/24/2018 8:03 PM 3.2: Log Functions
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Defining Logarithms Logarithms are defined as the INVERSE of an exponential function. It can be used specifically to find base powers. They are 10-fold. Exponential form: π π = π; where π is the BASE, π is the POWER, and π is the VALUE. Logarithmic form: π₯π¨π π π=π; π is the BASE, π is the ARGUMENT, and π is the POWER. It is read as βlog of π base πβ or βlog base π of πβ If the base is not given (such as log 3) it is understood to be COMMON BASE OF 10. 11/24/2018 8:03 PM 3.2: Log Functions
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Definition Logarithms are defined as the INVERSE of an exponential function. It can be used specifically to find base powers. They are 10-fold. If there is not a base given, the base is ALWAYS 10. 11/24/2018 8:03 PM 3.2: Log Functions
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Review What is the inverse function of π=ππ? 11/24/2018 8:03 PM
3.2: Log Functions
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To Identify Logarithms
π = Base π = Power/Argument π = Value 11/24/2018 8:03 PM 3.2: Log Functions
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The Snail 11/24/2018 8:03 PM 3.2: Log Functions
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= Example 1 Given π π =ππ, write this problem in logarithmic form
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Example 2 Given π₯π¨π π π ππ =βπ, write this problem in exponential form = 11/24/2018 8:03 PM 3.2: Log Functions
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Your Turn Given ππ βπ = π πππ , write this problem in logarithmic form
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Example 3 Given π₯π¨π ππ π=π, write this problem in exponential form and solve for π (without a calculator) 11/24/2018 8:03 PM 3.2: Log Functions
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Example 3 Given π₯π¨π ππ π=π, write this problem in exponential form and solve for π (without a calculator) 11/24/2018 8:03 PM 3.2: Log Functions
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Your Turn Given π₯π¨π π π , write this problem in exponential form and solve (without a calculator) 11/24/2018 8:03 PM 3.2: Log Functions
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Example 4 Given π₯π¨π β‘πππ=π, write this problem in exponential form and solve for π 11/24/2018 8:03 PM 3.2: Log Functions
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Your Turn Given π₯π¨π π ππππ = π, write this problem in exponential form and solve for π If there isnβt a base given, assume the base to beβ¦ 11/24/2018 8:03 PM 3.2: Log Functions
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Graphing Logs Exponential functions are its inverse
π₯π¨π π π is the parent function where it always passes π, π and π, π Domain: π, β , Range: ββ, β Equation: π=π π₯π¨π π π© πβπͺ +π« π¨ is the reflection over π-axis π is the base π© is the reflection over the π-axis πͺ = translation of the right or left π« = translation up or down 11/24/2018 8:03 PM 3.2: Log Functions
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Example 5 Graph π= π π and identify domain, range, and asymptote βπ βπ
π βπ = π π π = π π βπ π βπ = π π π = π π βπ π βπ = π π π π π = π π =π π π π =π π π π =π π π π =π π π βπ βπ βπ π π π π 11/24/2018 8:03 PM 3.2: Log Functions
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Example 6 Graph π= π₯π¨π π π and identify domain, range, and asymptote
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Example 6 Graph π= π₯π¨π π π and identify domain, range, and asymptote π
π= π₯π¨π π π π π π π βπ =π βπ π π π βπ =π βπ π π π βπ =π βπ π π π =π π π π π =π π π π =π π π π =π π π π= π₯π¨π π π π π₯π¨π π π =βπ βπ π₯π¨π π π =βπ βπ π₯π¨π π π =βπ βπ π₯π¨π π π =π π π₯π¨π π π =π π π₯π¨π π π =π π π₯π¨π π π =π π π π= π₯π¨π π π π βπ βπ βπ π π π π π π= π₯π¨π π π π π βπ =π βπ π βπ =π βπ π βπ =π βπ π π =π π π π =π π π π =π π π π =π π 11/24/2018 8:03 PM 3.2: Log Functions
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Your Turn Graph π= π₯π¨π π π and identify domain, range, and asymptote π
π= π₯π¨π π π π π ππ π βπ =π βπ π π π βπ =π βπ π π π βπ =π βπ π π π =π π π π π =π π π π =π π ππ π π =π 11/24/2018 8:03 PM 3.2: Log Functions
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Example 7 Graph π= π₯π¨π π πβπ and identify domain, range, and asymptote
π= π₯π¨π π π π π ππ π βπ =π βπ π π π βπ =π βπ π π π βπ =π βπ π π π =π π π π π =π π π π =π π ππ π π =π 11/24/2018 8:03 PM 3.2: Log Functions
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Example 7 Graph π= π₯π¨π π πβπ and identify domain, range, and asymptote
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Example 7 Graph π= π₯π¨π π πβπ and identify domain, range, and asymptote
π= π₯π¨π π π π π ππ π βπ =π βπ π π π βπ =π βπ π π π βπ =π βπ π π π =π π π π π =π π π π =π π ππ π π =π π π= π₯π¨π π π π ππ ππ π βπ =πβπ βπ ππ π π βπ =πβπ βπ π π π βπ =πβπ βπ π π π =πβπ π π π π =πβπ π ππ π π =πβπ ππ π π =πβπ π π π= π₯π¨π π π π π ππ +π π βπ =πβπ βπ π π +π π βπ =πβπ βπ π π +π π βπ =πβπ βπ π+π π π =πβπ π π+π π π =πβπ π π+π π π =πβπ π ππ+π π π =πβπ π 11/24/2018 8:03 PM 3.2: Log Functions
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Example 8 Graph π= π₯π¨π π π πβπ +π and identify domain, range, and asymptote π π= π₯π¨π π π π ππ ππ βπ π βπ =πβπ βπ+π ππ π βπ π βπ =πβπ βπ+π π π βπ π βπ =πβπ βπ+π πβπ π π =πβπ π+π πβπ π π =πβπ π+π ππβπ π π =πβπ π+π ππβπ π π =πβπ π+π π π= π₯π¨π π π π ππ ππ π βπ =πβπ π ππ π π βπ =πβπ π π π π βπ =πβπ π π π π =πβπ π π π π =πβπ ππ π π =πβπ π ππ π π =πβπ π π π= π₯π¨π π π π ππ ππ βπ π βπ =πβπ βπ ππ π βπ π βπ =πβπ βπ π π βπ π βπ =πβπ βπ πβπ π π =πβπ π πβπ π π =πβπ π ππβπ π π =πβπ π ππβπ π π =πβπ π 11/24/2018 8:03 PM 3.2: Log Functions
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Example 9 Graph π=βπ π₯π¨π π π+π βπ and identify domain, range, and asymptote 11/24/2018 8:03 PM 3.2: Log Functions
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Your Turn Graph π= π₯π¨π π β πβπ βπ and identify domain, range, and asymptote 11/24/2018 8:03 PM 3.2: Log Functions
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Assignment Worksheet 11/24/2018 8:03 PM 3.2: Log Functions
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