Identifying and Graphing Log Functions Section 3.2 Precalculus PreAP/Dual, Revised ©2017 viet.dang@humble.k12.tx.us 11/24/2018 8:03 PM 3.2: Log Functions

Real-Life Situation The pH scale is used in chemistry to determine the acidity or alkalinity of a solution. The scale ranges from 𝟏 to 𝟏𝟒, with 𝟏 being the most acidic and 𝟏𝟒 the most alkaline The difference in strength of an acid of 𝒑𝑯 𝟏 and that of 𝒑𝑯 𝟐 is not twofold, but tenfold The pH scale is actually a logarithm in the form: 𝒍𝒐𝒈𝟏𝟎, thus, 𝒑𝑯 𝟏 = 𝒍𝒐𝒈𝟏𝟎𝟏𝟎, and 𝒑𝑯 𝟐 = 𝒍𝒐𝒈𝟏𝟎𝟏𝟎𝟎 11/24/2018 8:03 PM 3.2: Log Functions

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

Real-Life Situation The Modified Richter Scale uses a modified scale. It is not ten-fold A separate equation is used Music “Semitones” The interval between two notes in semitones is the base-21/12 logarithm of the frequency ratio (or equivalently, 12 times the base-2 logarithm). Astronomy The magnitude measures the stars’ brightness logarithmically with vision Source: Wikipedia 11/24/2018 8:03 PM 3.2: Log Functions

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: 𝒃 𝒙 = 𝒂; where 𝒃 is the BASE, 𝒙 is the POWER, and 𝒂 is the VALUE. Logarithmic form: 𝐥𝐨𝐠 𝒃 𝒂=𝒙; 𝒃 is the BASE, 𝒂 is the ARGUMENT, and 𝒙 is the POWER. It is read as “log of 𝒂 base 𝒃” or “log base 𝒃 of 𝒂” 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

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

Review What is the inverse function of 𝒚=𝟐𝒙? 11/24/2018 8:03 PM 3.2: Log Functions

To Identify Logarithms 𝒃 = Base 𝒙 = Power/Argument 𝒂 = Value 11/24/2018 8:03 PM 3.2: Log Functions

The Snail 11/24/2018 8:03 PM 3.2: Log Functions

= Example 1 Given 𝟐 𝟒 =𝟏𝟔, write this problem in logarithmic form 11/24/2018 8:03 PM 3.2: Log Functions

Example 2 Given 𝐥𝐨𝐠 𝟒 𝟏 𝟏𝟔 =−𝟐, write this problem in exponential form = 11/24/2018 8:03 PM 3.2: Log Functions

Your Turn Given 𝟏𝟎 −𝟐 = 𝟏 𝟏𝟎𝟎 , write this problem in logarithmic form 11/24/2018 8:03 PM 3.2: Log Functions

Example 3 Given 𝐥𝐨𝐠 𝟐𝟕 𝟗=𝒙, write this problem in exponential form and solve for 𝒙 (without a calculator) 11/24/2018 8:03 PM 3.2: Log Functions

Example 3 Given 𝐥𝐨𝐠 𝟐𝟕 𝟗=𝒙, write this problem in exponential form and solve for 𝒙 (without a calculator) 11/24/2018 8:03 PM 3.2: Log Functions

Your Turn Given 𝐥𝐨𝐠 𝟖 𝟖 , write this problem in exponential form and solve (without a calculator) 11/24/2018 8:03 PM 3.2: Log Functions

Example 4 Given 𝐥𝐨𝐠⁡𝟏𝟎𝟎=𝒙, write this problem in exponential form and solve for 𝒙 11/24/2018 8:03 PM 3.2: Log Functions

Your Turn Given 𝐥𝐨𝐠 𝟏 𝟏𝟎𝟎𝟎 = 𝒙, write this problem in exponential form and solve for 𝒙 If there isn’t a base given, assume the base to be… 11/24/2018 8:03 PM 3.2: Log Functions

Graphing Logs Exponential functions are its inverse 𝐥𝐨𝐠 𝒃 𝒙 is the parent function where it always passes 𝟏, 𝟎 and 𝒃, 𝟏 Domain: 𝟎, ∞ , Range: −∞, ∞ Equation: 𝒚=𝒂 𝐥𝐨𝐠 𝒃 𝑩 𝒙−𝑪 +𝑫 𝑨 is the reflection over 𝒙-axis 𝒃 is the base 𝑩 is the reflection over the 𝒚-axis 𝑪 = translation of the right or left 𝑫 = translation up or down 11/24/2018 8:03 PM 3.2: Log Functions

Example 5 Graph 𝒚= 𝟐 𝒙 and identify domain, range, and asymptote −𝟑 −𝟐 𝟐 −𝟑 = 𝟏 𝟐 𝟑 = 𝟏 𝟖 −𝟐 𝟐 −𝟐 = 𝟏 𝟐 𝟐 = 𝟏 𝟒 −𝟏 𝟐 −𝟏 = 𝟏 𝟐 𝟎 𝟐 𝟎 = 𝟏 𝟏 =𝟏 𝟏 𝟐 𝟏 =𝟐 𝟐 𝟐 𝟐 =𝟒 𝟑 𝟐 𝟑 =𝟖 𝒙 𝒚 −𝟑 −𝟐 −𝟏 𝟎 𝟏 𝟐 𝟑 11/24/2018 8:03 PM 3.2: Log Functions

Example 6 Graph 𝒚= 𝐥𝐨𝐠 𝟐 𝒙 and identify domain, range, and asymptote 11/24/2018 8:03 PM 3.2: Log Functions

Example 6 Graph 𝒚= 𝐥𝐨𝐠 𝟐 𝒙 and identify domain, range, and asymptote 𝒙 𝒚= 𝐥𝐨𝐠 𝟐 𝒙 𝒚 𝟏 𝟖 𝟐 −𝟑 =𝒙 −𝟑 𝟏 𝟒 𝟐 −𝟐 =𝒙 −𝟐 𝟏 𝟐 𝟐 −𝟏 =𝒙 −𝟏 𝟏 𝟐 𝟎 =𝒙 𝟎 𝟐 𝟐 𝟏 =𝒙 𝟒 𝟐 𝟐 =𝒙 𝟖 𝟐 𝟑 =𝒙 𝟑 𝒙 𝒚= 𝐥𝐨𝐠 𝟐 𝒙 𝒚 𝐥𝐨𝐠 𝟐 𝒙 =−𝟑 −𝟑 𝐥𝐨𝐠 𝟐 𝒙 =−𝟐 −𝟐 𝐥𝐨𝐠 𝟐 𝒙 =−𝟏 −𝟏 𝐥𝐨𝐠 𝟐 𝒙 =𝟎 𝟎 𝐥𝐨𝐠 𝟐 𝒙 =𝟏 𝟏 𝐥𝐨𝐠 𝟐 𝒙 =𝟐 𝟐 𝐥𝐨𝐠 𝟐 𝒙 =𝟑 𝟑 𝒙 𝒚= 𝐥𝐨𝐠 𝟐 𝒙 𝒚 −𝟑 −𝟐 −𝟏 𝟎 𝟏 𝟐 𝟑 𝒙 𝒚= 𝐥𝐨𝐠 𝟐 𝒙 𝒚 𝟐 −𝟑 =𝒙 −𝟑 𝟐 −𝟐 =𝒙 −𝟐 𝟐 −𝟏 =𝒙 −𝟏 𝟐 𝟎 =𝒙 𝟎 𝟐 𝟏 =𝒙 𝟏 𝟐 𝟐 =𝒙 𝟐 𝟐 𝟑 =𝒙 𝟑 11/24/2018 8:03 PM 3.2: Log Functions

Your Turn Graph 𝒚= 𝐥𝐨𝐠 𝟑 𝒙 and identify domain, range, and asymptote 𝒙 𝒚= 𝐥𝐨𝐠 𝟑 𝒙 𝒚 𝟏 𝟐𝟕 𝟑 −𝟑 =𝒙 −𝟑 𝟏 𝟗 𝟑 −𝟐 =𝒙 −𝟐 𝟏 𝟑 𝟑 −𝟏 =𝒙 −𝟏 𝟏 𝟑 𝟎 =𝒙 𝟎 𝟑 𝟑 𝟏 =𝒙 𝟗 𝟑 𝟐 =𝒙 𝟐 𝟐𝟕 𝟑 𝟑 =𝒙 11/24/2018 8:03 PM 3.2: Log Functions

Example 7 Graph 𝒚= 𝐥𝐨𝐠 𝟑 𝒙−𝟏 and identify domain, range, and asymptote 𝒚= 𝐥𝐨𝐠 𝟑 𝒙 𝒚 𝟏 𝟐𝟕 𝟑 −𝟑 =𝒙 −𝟑 𝟏 𝟗 𝟑 −𝟐 =𝒙 −𝟐 𝟏 𝟑 𝟑 −𝟏 =𝒙 −𝟏 𝟏 𝟑 𝟎 =𝒙 𝟎 𝟑 𝟑 𝟏 =𝒙 𝟗 𝟑 𝟐 =𝒙 𝟐 𝟐𝟕 𝟑 𝟑 =𝒙 11/24/2018 8:03 PM 3.2: Log Functions

Example 7 Graph 𝒚= 𝐥𝐨𝐠 𝟑 𝒙−𝟏 and identify domain, range, and asymptote 11/24/2018 8:03 PM 3.2: Log Functions

Example 7 Graph 𝒚= 𝐥𝐨𝐠 𝟑 𝒙−𝟏 and identify domain, range, and asymptote 𝒚= 𝐥𝐨𝐠 𝟑 𝒙 𝒚 𝟏 𝟐𝟕 𝟑 −𝟑 =𝒙 −𝟑 𝟏 𝟗 𝟑 −𝟐 =𝒙 −𝟐 𝟏 𝟑 𝟑 −𝟏 =𝒙 −𝟏 𝟏 𝟑 𝟎 =𝒙 𝟎 𝟑 𝟑 𝟏 =𝒙 𝟗 𝟑 𝟐 =𝒙 𝟐 𝟐𝟕 𝟑 𝟑 =𝒙 𝒙 𝒚= 𝐥𝐨𝐠 𝟑 𝒙 𝒚 𝟐𝟖 𝟐𝟕 𝟑 −𝟑 =𝒙−𝟏 −𝟑 𝟏𝟎 𝟗 𝟑 −𝟐 =𝒙−𝟏 −𝟐 𝟒 𝟑 𝟑 −𝟏 =𝒙−𝟏 −𝟏 𝟐 𝟑 𝟎 =𝒙−𝟏 𝟎 𝟒 𝟑 𝟏 =𝒙−𝟏 𝟏 𝟏𝟎 𝟑 𝟐 =𝒙−𝟏 𝟐𝟖 𝟑 𝟑 =𝒙−𝟏 𝟑 𝒙 𝒚= 𝐥𝐨𝐠 𝟑 𝒙 𝒚 𝟏 𝟐𝟕 +𝟏 𝟑 −𝟑 =𝒙−𝟏 −𝟑 𝟏 𝟗 +𝟏 𝟑 −𝟐 =𝒙−𝟏 −𝟐 𝟏 𝟑 +𝟏 𝟑 −𝟏 =𝒙−𝟏 −𝟏 𝟏+𝟏 𝟑 𝟎 =𝒙−𝟏 𝟎 𝟑+𝟏 𝟑 𝟏 =𝒙−𝟏 𝟏 𝟗+𝟏 𝟑 𝟐 =𝒙−𝟏 𝟐 𝟐𝟕+𝟏 𝟑 𝟑 =𝒙−𝟏 𝟑 11/24/2018 8:03 PM 3.2: Log Functions

Example 8 Graph 𝒚= 𝐥𝐨𝐠 𝟑 𝟐 𝒙−𝟏 +𝟑 and identify domain, range, and asymptote 𝒙 𝒚= 𝐥𝐨𝐠 𝟑 𝒙 𝒚 𝟐𝟖 𝟐𝟕 ∙𝟐 𝟑 −𝟑 =𝒙−𝟏 −𝟑+𝟑 𝟏𝟎 𝟗 ∙𝟐 𝟑 −𝟐 =𝒙−𝟏 −𝟐+𝟑 𝟒 𝟑 ∙𝟐 𝟑 −𝟏 =𝒙−𝟏 −𝟏+𝟑 𝟐∙𝟐 𝟑 𝟎 =𝒙−𝟏 𝟎+𝟑 𝟒∙𝟐 𝟑 𝟏 =𝒙−𝟏 𝟏+𝟑 𝟏𝟎∙𝟐 𝟑 𝟐 =𝒙−𝟏 𝟐+𝟑 𝟐𝟖∙𝟐 𝟑 𝟑 =𝒙−𝟏 𝟑+𝟑 𝒙 𝒚= 𝐥𝐨𝐠 𝟑 𝒙 𝒚 𝟓𝟔 𝟐𝟕 𝟑 −𝟑 =𝒙−𝟏 𝟎 𝟐𝟎 𝟗 𝟑 −𝟐 =𝒙−𝟏 𝟏 𝟖 𝟑 𝟑 −𝟏 =𝒙−𝟏 𝟐 𝟒 𝟑 𝟎 =𝒙−𝟏 𝟑 𝟖 𝟑 𝟏 =𝒙−𝟏 𝟐𝟎 𝟑 𝟐 =𝒙−𝟏 𝟓 𝟓𝟔 𝟑 𝟑 =𝒙−𝟏 𝟔 𝒙 𝒚= 𝐥𝐨𝐠 𝟑 𝒙 𝒚 𝟐𝟖 𝟐𝟕 ∙𝟐 𝟑 −𝟑 =𝒙−𝟏 −𝟑 𝟏𝟎 𝟗 ∙𝟐 𝟑 −𝟐 =𝒙−𝟏 −𝟐 𝟒 𝟑 ∙𝟐 𝟑 −𝟏 =𝒙−𝟏 −𝟏 𝟐∙𝟐 𝟑 𝟎 =𝒙−𝟏 𝟎 𝟒∙𝟐 𝟑 𝟏 =𝒙−𝟏 𝟏 𝟏𝟎∙𝟐 𝟑 𝟐 =𝒙−𝟏 𝟐 𝟐𝟖∙𝟐 𝟑 𝟑 =𝒙−𝟏 𝟑 11/24/2018 8:03 PM 3.2: Log Functions

Example 9 Graph 𝒚=−𝟐 𝐥𝐨𝐠 𝟐 𝒙+𝟐 −𝟑 and identify domain, range, and asymptote 11/24/2018 8:03 PM 3.2: Log Functions

Your Turn Graph 𝒚= 𝐥𝐨𝐠 𝟐 − 𝒙−𝟒 −𝟑 and identify domain, range, and asymptote 11/24/2018 8:03 PM 3.2: Log Functions

Assignment Worksheet 11/24/2018 8:03 PM 3.2: Log Functions